Room acoustics simulations using FDTD method · Boundary and medium modelling using compact ﬁnite diﬀerence schemes in simulations of room acoustics for audio and architectural

Boundary and medium modelling using compact

finite difference schemes in simulations of room

acoustics for audio and architectural design

applications

Konrad KowalczykB.Eng., M.Sc.

Sonic Arts Research Centre

School of Electronics, Electrical Engineering and Computer Science

Queen’s University Belfast

Submitted for the Degree of Doctor of Philosophy

November 2008

To Kasia...

Abstract

Simulation of acoustic spaces with the aim of developing virtual immersive applications

and architectural design applications is one of the key areas in the field of audio signal

processing. In this thesis, a complete method for simulating room acoustics using compact

finite difference time domain (FDTD) schemes is presented.

A family of compact explicit and implicit schemes approximating the wave equation is

analysed in terms of stability, accuracy, and computational efficiency. The most accurate

and isotropic schemes based on a rectilinear nonstaggered grid are identified, and the

optimally efficient explicit schemes are indicated.

Novel FDTD formulations of frequency-independent and frequency-dependent bound-

aries of a locally reacting surface type are proposed, including a full treatment of corners

and boundary edges. In particular, it is proposed to model generally frequency-dependent

boundaries by local incorporation of a digital impedance filter (DIF), and the resulting

formulae for compact explicit schemes are provided. In addition, a numerical boundary

analysis (NBA) procedure is proposed as a technique for analytic evaluation of the numer-

ical reflectance of the presented boundary models. The digital impedance filter model is

also extended to model controllable surface diffusion based on the concept of phase grating

diffusers.

Results obtained from numerical experiments and numerical boundary analysis confirm

the high accuracy of the proposed boundary models, the reflectance of which is shown

to closely approximate locally reacting surface theory for different angles of incidence

and various impedances. Furthermore, the results indicate that boundary formulations

based on the identified accurate and isotropic schemes are also very accurate in terms of

numerical reflectance, and outperform directly related methods such as Yee’s scheme and

the standard digital waveguide mesh. In addition, one particular scheme - referred to as

the interpolated wideband scheme - is suggested as the best FDTD scheme for most audio

applications.

i

Acknowledgements

This research has been carried out at the Sonic Arts Research Centre, Queen’s University

Belfast between October 2005 and December 2008. I would like to express my gratitude to

my supervisor Dr. Maarten van Walstijn for letting me pursue a Ph.D. in this topic and

organising financial support for my research. I am deeply grateful for his dedicated and

consistent support, guidance, and patience in teaching me technical writing. Our weekly

discussions and his open-door policy greatly helped in the successful completion of this

thesis.

Many thanks go to my colleagues from SARC, both postgraduate students and staff,

for creating a very vibrant and inspiring environment, with a family-like atmosphere.

There are so many of you that it makes it impossible to mention you all here. I am

particularly grateful for your friendship in and outside SARC, and the very best social life

which successfully provided me with numerous pleasant distractions from this work; this

also includes late night jam sessions and Sunday football games.

I am thankful to Dr. Stefan Bilbao for his continuous interest in this work, many

helpful suggestions and personalised lectures introducing me to the concept of FDTD

methods. Very special thanks to Prof. Roger Woods from Queen’s who has provided

invaluable guidance and support throughout my graduate career.

I have had the great honour of being a visiting Ph.D. student at the Center for Com-

puter Research in Music and Acoustics (CCRMA), Stanford University, in 2007 and the

Audio Lab, Department of Electronics, University of York, in 2008. I am greatly indebted

to Prof. Julius O. Smith for a very inspirational stay at CCRMA, for many insightful dis-

cussions and for finding time to discuss my work despite the busy schedule. Many thanks

to Dr. Damian Murphy for warmly hosting me in York leading to a fruitful collaboration,

and for our highly interesting lunch conversations.

I would like to thank Prof. Peter Svensson, Patty Huang, Vasileios Chatziioannou, Dr.

Tapio Lokki, Prof. Rudolf Rabenstein, and Prof. Diemer de Vries for insightful discussions

related to my work on numerous occasions.

Sincere thanks go to my girlfriend and soon to become wife, Kasia, for her love, con-

tinuous encouragement and sharing ups and downs related to the Ph.D. experience. Last

iii

but definitely not least, I am indebted to my parents, sister Ela and all my friends, for

making me who I am and for being there for me.

I would also like to thank my football coach for giving me a place in the first team and

trusting my scoring skills, even when it seemed to me almost impossible to score a goal.

The financial support of the European Social Fund is acknowledged.

Contents

Abstract i

Acknowledgements iii

1 Introduction 1

1.1 Research Objectives and Applications . . . . . . . . . . . . . . . . . . . . . 3

1.2 Room Acoustics Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Related Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Fundamentals of Room Acoustics 11

2.1 Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Sound Pressure Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Locally Reacting Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.1 Boundary Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.2 Reflection at Normal Incidence . . . . . . . . . . . . . . . . . . . . . 15

2.3.3 Reflection at Oblique Incidence . . . . . . . . . . . . . . . . . . . . . 17

2.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Eigenmode Model (for Rigid Walls) . . . . . . . . . . . . . . . . . . . . . . . 19

2.5 Acoustical Porous Material . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6 Diffusers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.6.1 Maximum Length Sequence . . . . . . . . . . . . . . . . . . . . . . . 22

2.6.2 Quadratic Residue Diffuser . . . . . . . . . . . . . . . . . . . . . . . 23

2.6.3 Modulated Quadratic Residue Diffuser . . . . . . . . . . . . . . . . . 25

2.6.4 Diffractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.6.5 Curved Diffusers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.6.6 Fractional Brownian Diffusers . . . . . . . . . . . . . . . . . . . . . . 28

2.7 Diffusion Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

vi

2.7.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.7.2 Scattering Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Elements of Numerical Modelling 33

3.1 Room Acoustics Modelling Methods . . . . . . . . . . . . . . . . . . . . . . 34

3.1.1 Geometrical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1.2 Wave-based Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.1.3 Motivation for the Chosen Method . . . . . . . . . . . . . . . . . . . 37

3.2 The Finite Difference Time Domain Method . . . . . . . . . . . . . . . . . . 38

3.2.1 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.2 Dispersion Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2.3 Staggered FDTD Method . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2.4 Digital Waveguide Mesh . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2.5 Implicit Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.3 Frequency Warping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.4 Solving Tridiagonal Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.5 Fractional Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4 Compact FDTD Schemes 68

4.1 2D Compact FDTD Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.1.1 Special Cases of Explicit Schemes . . . . . . . . . . . . . . . . . . . . 70

4.1.2 Special cases of implicit schemes . . . . . . . . . . . . . . . . . . . . 72

4.1.3 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.1.4 Numerical dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.1.5 Dispersion analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.1.6 Accuracy and Isotropy . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.1.7 Computational Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.2 3D Compact FDTD Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.2.1 Special Cases of 3D Explicit Schemes . . . . . . . . . . . . . . . . . 88

4.2.2 3D Compact Implicit Schemes . . . . . . . . . . . . . . . . . . . . . 92

4.2.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.2.4 Numerical Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.2.5 Dispersion Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.2.6 Accuracy and Isotropy . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.2.7 Computational Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

vii

5 FDTD Formulation of Locally Reacting Surfaces 103

5.1 Locally Reacting Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.2 Frequency-independent Boundaries . . . . . . . . . . . . . . . . . . . . . . . 106

5.2.1 2D Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.2.2 1D Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.2.3 Corners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.3 Frequency-dependent Boundaries . . . . . . . . . . . . . . . . . . . . . . . . 109

5.3.1 Boundary Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.3.2 Corners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.4 Boundaries in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113


5.4.2 Corners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.5 Numerical Boundary Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.5.1 2D boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.5.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.5.3 3D Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.6.1 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.6.2 2D Frequency-independent Boundary . . . . . . . . . . . . . . . . . 123

5.6.3 2D Frequency-dependent Boundaries . . . . . . . . . . . . . . . . . . 126

5.6.4 3D Boundary Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6 Modelling Frequency-Dependent Boundaries as Digital Impedance Fil-

ters 133

6.1 Digital Impedance Filter (DIF) . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.2 2D DIF Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135


6.2.2 Other Rectilinear-grid Boundaries . . . . . . . . . . . . . . . . . . . 139

6.2.3 K-DWM Implementation . . . . . . . . . . . . . . . . . . . . . . . . 140

6.2.4 Corners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.3 3D DIF Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141


6.3.2 Corners and Boundary Edges . . . . . . . . . . . . . . . . . . . . . . 142


6.4.1 2D DIF Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.4.2 3D DIF Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.5 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

viii

6.5.1 1D Boundary Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.5.2 Impedance Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.5.3 Results of the 2D DIF Model . . . . . . . . . . . . . . . . . . . . . . 148

6.5.4 Results of the 3D DIF Model . . . . . . . . . . . . . . . . . . . . . . 153

6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7 Compact Explicit Formulation of the DIF Model 158

7.1 Compact Explicit DIF Formulation . . . . . . . . . . . . . . . . . . . . . . . 159

7.1.1 2D Compact Explicit DIF Boundary Model . . . . . . . . . . . . . . 159

7.1.2 2D Compact Explicit DIF Corners . . . . . . . . . . . . . . . . . . . 163

7.1.3 3D Compact Explicit DIF Boundary Model . . . . . . . . . . . . . . 165

7.1.4 3D Compact Explicit DIF Corners . . . . . . . . . . . . . . . . . . . 168


7.3 2D Boundary Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

7.3.1 Frequency-independent Results . . . . . . . . . . . . . . . . . . . . . 174

7.3.2 Frequency-dependent Results . . . . . . . . . . . . . . . . . . . . . . 176

7.4 3D Boundary Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

7.4.1 Frequency-independent Results . . . . . . . . . . . . . . . . . . . . . 178

7.4.2 Frequency-dependent Results . . . . . . . . . . . . . . . . . . . . . . 179

7.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

8 A Phase Grating Approach to Modelling Surface Diffusion 186

8.1 A Method for Simulating Diffusive Surfaces . . . . . . . . . . . . . . . . . . 187

8.1.1 A Phase Grating Approach . . . . . . . . . . . . . . . . . . . . . . . 187

8.1.2 Relationship between the Well Depth and Delay Length . . . . . . . 188

8.1.3 Fractional Delay Implementation . . . . . . . . . . . . . . . . . . . . 189

8.1.4 Diffusion Parameter Control . . . . . . . . . . . . . . . . . . . . . . . 189

8.2 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

8.2.1 Test Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

8.2.2 Modelled Diffusers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

8.2.3 Frequency-domain Results . . . . . . . . . . . . . . . . . . . . . . . . 194

8.2.4 Time-domain Results . . . . . . . . . . . . . . . . . . . . . . . . . . 201

8.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

9 Conclusions and Recommendations 205

9.1 General Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

9.2 Future Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

ix

Bibliography 210

1

Chapter 1

Introduction

Over the past two decades, various computer modelling techniques have been developed

for auralisation purposes. With the rise of the role of the audio part in many multimedia

applications, computational modelling of acoustic spaces has recently gained wider inter-

est. The level of accuracy to which the sound environment is modelled depends strongly

on a particular application and availability of computational resources for audio signal

processing. In the simplest case of real-time simulations in interactive multimedia appli-

cations and computer games, usually only the sound sources are rendered, leaving out the

acoustic effects of the surrounding environment. On the other hand, due to an increased

need for realism, many applications of computer-based modelling of room acoustics require

more details to be simulated. Previously, such more accurate and computationally expen-

sive modelling techniques were utilised in the creation of naturally sounding reverberation

units and room acoustics prediction for architectural design applications. However, due

to the increase in the computational power of commonly available processors, more de-

tailed modelling of sound propagation in acoustic spaces can now be integrated in generic

entertainment and multimedia applications.

The rapid development of virtual reality applications and multimedia technology has

stimulated the development and inclusion of acoustic modelling in numerous applications.

Therefore, there is a need for algorithms enabling the creation of virtual acoustic environ-

ments with multiple moving sound sources, which can be freely explored by the listener

or a group of listeners. The key feature of such systems is their perceptual immersiveness,

which can be defined as a feeling of realism experienced in a virtual acoustic space. Thus

the realistic quality of sound should be ensured, which can be obtained with perceptual

or physical approaches. With a perceptual approach, a plausible sound field is generated

using perceptual parameters. However, since the acoustics of the virtual space is not ex-

plicitly modelled, it is not suitable for room acoustics prediction. The main applications

include computer games, the creation of spatial effects for composers and plausible rever-

Chapter 1. Introduction 2

Sound source

modelling

Room acoustics

modelling

Receiver modelling according

to reproduction system

Virtual acoustic

application

Figure 1.1: Auralisation stages.

beration units in music production. The perceptual approach is also applied in the Spat

software [48] and for sound environment modelling in MPEG-4 scene description language

[117]. Conversely, the physical approach is based on modelling the acoustics of the virtual

enclosed space defined by physical parameters such as room shape and boundary material.

Consequently, it can be used to predict the acoustics of auditoria in architectural design

and are generally applicable to multimedia applications. Some example applications of

a physical approach include ODEON [78] software for room acoustics and virtual reality

applications such as DIVA [88] and [70].

The most popular approach to auralisation consists in computing one or more room

impulse responses of the modelled space and convolving them with a dry source signal.

The impulse responses are captured at a receiver position in a format defined by the

reproduction technique, and next the soundfield of the simulated acoustic space is repro-

duced in a listening environment [110]. Consequently, we can distinguish three modelling

components in an auralisation system, namely the source, room acoustics and listener, as

illustrated in Figure 1.1.

For a source of sound, the spatial localisation and directivity should be modelled.

Dry and monophonic input signal can then be fed into the system as a pre-recorded or

synthesised sound. Receiver modelling refers to the position and directivity of a listener

using available reproduction systems. For that purpose, various sound reproduction tech-

niques are available, including binaural techniques [74] for sound field reproduction for a

single listener using head-related transfer functions (HRTF) and multichannel loudspeak-

ers techniques. The latter have the advantage that a listener can freely move the head

without compromising the reproduced quality and include wavefield synthesis (WFS) [9],

Ambisonics [41] and vector-base amplitude-panning (VBAP) [83] techniques. Simulation

of the room acoustics is the main component of the modelling structure.


1.1 Research Objectives and Applications

The main goal of this research is to develop improved methods for the simulation of sound

propagation in acoustic spaces for architectural design and audio applications. Perceptual

realism and a high level of accuracy are increasingly required in these applications. This

research aims to develop numerical algorithms that are applicable to creating an immersive

acoustic environment, which allows the simulation of acoustic spaces of a complex shape

with multiple moving sound sources and listeners.

Computational modelling of acoustic spaces is fundamental for various auralisation

and room acoustics applications. Possible applications of room acoustic modelling are

architectural design software and the analysis and evaluation of existing acoustic spaces.

Non-real-time high accuracy simulations can be used to predict the soundfield in music

performance spaces, recording studio design, and for architectural design purposes. Such

a numerical tool is beneficial in the process of designing spaces with desirable acoustics, as

it enables predicting the performance before constructing the building. Similarly, a high-

accuracy simulation would be very beneficial in early stages of diffuser design, where sound

scattering from the boundary surface in time and space domain could be investigated. An

accurate acoustic model could also be applied to modelling of complex loudspeaker systems

with the emphasis on source directivity.

The methods developed for room acoustics could also be applied to virtual sound

environments and to the creation of spatial sound effects for multimedia applications.

Plausible sound field modelling could be applicable in naturally sounding reverberation

units. A good example of a multimedia application is the creation of a realistic reverberant

soundtrack for an animation or film.

In order to meet the aforementioned objectives, it is necessary to address the issues of

sound source modelling, acoustic space simulation and receiver modelling according to the

reproduction technique. However, in this thesis we focus entirely on room acoustics mod-

elling which constitutes the main modelling component. Sound propagation in an acoustic

space, room geometry, reflections at boundaries, occlusion, wave interference and diffusion

are key features that need to be reproduced in order to reach a high level of accuracy.

These objectives can be achieved with the use of numerical simulation techniques, which

is the main field of the undertaken research.

1.2 Room Acoustics Modelling

Research into numerical simulation of acoustic spaces is dominated by two distinct ap-

proaches, namely the geometrical and the wave-based approach. The former is based on

soundfield decomposition and is computationally relatively efficient. ODEON is a good


example of calculating impulse responses based on the geometrical approach [78]; it uses

the image source method [6] for early reflection modelling and the ray tracing technique for

modelling diffusive reverberation. Similar hybrid approaches are also described in the con-

text of virtual acoustics and auralisation in [85, 70]. The aim of DIVA (Digital Interactive

Virtual Acoustics) project is to create a real-time environment for full audiovisual experi-

ence [88]. In this case, the image source method is employed for modelling up to six early

reflections, while the late reverberation is generated using recursive filter structures to al-

low real-time processing. Although soundfield decomposition methods are efficient, their

formulation is not entirely physical, and consequently their predictive capacity is rather

limited. This limitation is generally apparent for low and middle frequency ranges, and

particularly so when applied to modelling small enclosures or rooms with highly nonrigid

walls [123, 110].

On the other hand, wave-based methods simulate the acoustical equations directly

and therefore have the advantage of inherently modelling wave-related phenomena such

as diffraction, be it that the computational costs for wideband applications are high,

especially for modelling and auralisation of 3D spaces. The past few years have seen a

rise of interest in wave-based methods, partly driven by the steady increase of commonly

available processing power. These methods include finite difference time-domain (FDTD)

methods, digital waveguide mesh (DWM) modelling, the finite element method (FEM), the

boundary element method (BEM), and the functional transform method (FTM). Wave-

based techniques have the advantage of modelling acoustic spaces with great detail, which

results in highly accurate simulations. However, this is achieved at the expense of the

computational cost, which rises exponentially with increasing sampling frequency. It is

therefore important to formulate these models as efficient as possible.

This research focuses on FDTD modelling, which is a good choice for virtual acoustic

applications for the following reasons. Firstly, a wide body of knowledge and methods has

been developed since the 1960s in the field of electro-dynamics, the underlying equations of

which are identical to those of acoustic systems. Secondly, unlike finite element methods,

FDTD methods tend to use uniform grids, which are more suited to auralisation of virtual

spaces with moving sources and receivers. Note that in general irregular grid spacing

causes undesirable filtering effects. Finally, the formulation and implementation of FDTD

models is relatively straight-forward in comparison to some of the other approaches.

Since a substantially higher accuracy of wave-based methods is in general offset by a

much higher computational cost than for geometrical methods, hybrid approaches are con-

venient to address the problem of fine-tuning the balance between accuracy and efficiency

[110]. Such hybrids generally rely on a combination of a rigorous numerical technique

such as the FDTD method and a computationally efficient geometrical method for high

frequencies, examples of which can be found in [43, 62, 76]. In the long term, one may


expect that the burden of computational costs will be lessened by the growth in commonly

available processing power, where the development of multicore processors could be the

key step forward in bringing rigorous simulations of large 3D spaces within reach.

The main focus of this thesis is on modelling the acoustics of enclosed spaces, and

hence the main issues that should be addressed include modelling sound wave propaga-

tion, models of generally frequency-dependent boundaries, and modelling surface diffusion.

Note that other wave-related phenomena are inherently incorporated in the FDTD tech-

nique. Since the aim is to model multiple moving sound sources and listeners, the use of

off-line post-processing techniques such as frequency warping is excluded. Consequently,

throughout this thesis, we concentrate primarily on FDTD schemes with significantly re-

duced dispersion error without introducing much increase in computational cost, which

could be applied to on-line simulations. Furthermore, the considerations are constrained

to compact schemes on a rectilinear topology since it allows a straightforward fit of the

grid to rooms with parallel walls, which are dominant in real architecture.

The problem of developing accurate formulations of boundaries is an essential ingre-

dient in creating realistic and predictive FDTD simulations, especially given that realistic

boundaries are generally frequency-dependent. Strictly speaking, complete physical mod-

els of boundaries should include the transmission of waves in the wall. However, simulation

results in previous studies [15, 14] have suggested that in many practical cases there is no

significant difference if wave propagation in the wall is neglected. Therefore, in this thesis

it is assumed that any room surfaces are locally reacting, i.e. the reflective properties of

any point on the wall are completely characterised by a local impedance. Such generally

frequency-dependent boundary models with ensured scheme consistency between the room

interior and the boundary have a clear advantage that analytic prediction techniques can

be applied and the stability of the whole simulation is always guaranteed.

For auralisation purposes, strong simulation predictivity is in some cases of lesser

importance, the main objective shifting to enabling good control over the properties of

the simulated space, such as the overall room diffusivity. In the latter context, methods

for modelling controllable surface diffusion are required.

The issue of modelling directional sound sources and converting the output of the finite

difference grid according to various reproduction formats are not dealt with in this study.

Some interesting solutions to exciting the mesh include implementing transparent sources

in the FDTD grids [97] and modelling frequency-dependent directivity of sources in the

closely related digital waveguide mesh [45]. As far as receiver modelling is concerned, for

the low frequency range only, capturing pressure waves in points near the positions of a

listener’s ears should be sufficient [110]. A more general approach is based on plane-wave

decomposition which can be post-processed for the most of reproduction systems, but it

is computationally heavy [108]. Therefore, simple solutions for a specified reproduction


technique might be provide a useful balance, such as capturing B-format channels [107].

1.3 Thesis Overview

This thesis is divided in two major parts, namely the background information that can

be found in the literature and the author’s original contributions. An overview of the

basics of acoustics in enclosed spaces and the review of numerical techniques that are

applied in chapters to follow are presented in the first two chapters of this thesis. These

constitute a theoretical foundation for the work presented thereafter. The subsequent

chapters constitute the contributions to the field of FDTD modelling of room acoustics.

In Chapter 2, the fundamentals of acoustics related to sound propagation in enclosed

spaces are briefly reviewed, with the main focus on the properties of medium and bound-

aries. Basic acoustic laws are reviewed and the concept of locally reacting surfaces is dis-

cussed. In addition, this chapter provides an extensive overview of commercially available

diffusers and the standardised technique to measure the diffusivity of boundary surfaces.

Chapter 3 provides an overview of a number of numerical modelling techniques and

highlights the equivalence of various approaches. Methods based on the geometrical and

wave-based approaches are briefly discussed, and a more detailed motivation for the choice

of the FDTD method is provided. A number of techniques that are considered a subclass of

FDTD methods are reviewed, namely the digital waveguide mesh and the family compact

schemes based on staggered and nonstaggered grids. In particular, the analysis of stability

and dispersion of such methods is used to define the equivalence of these approaches.

For each technique, a short review of the boundary models available in the literature is

also provided. In addition, a compact implicit technique is discussed and the issue of a

computationally efficient implementation using the alternating direction implicit technique

is addressed.

Chapter 4 deals with modelling sound wave propagation in air, also referred to as

medium modelling. The family of compact finite difference time domain schemes based

on a nonstaggered rectilinear grid for approximating the 2D and 3D wave equation is

discussed. The issues of stability, accuracy and computational efficiency are investigated

for numerous special cases of a wide family of compact explicit and implicit schemes. The

presented analysis covers a wide range of techniques commonly used in the context of audio

such as the rectilinear digital waveguide mesh, the interpolated digital waveguide mesh,

and FDTD schemes such as the standard leapfrog, the octahedral, and the tetrahedral

schemes. As an alternative to these explicit techniques, the use of a fourth-order accurate

compact implicit finite difference technique is proposed for simulations in which very

high accuracy is required. The compact implicit formulation is presented for 2D and 3D

cases, including the most efficient splitting formulae for the alternating direction implicit


implementation. Compact explicit and implicit FDTD schemes are compared in terms of

numerical dispersion error, valid frequency ranges for accuracy and isotropy, computational

cost and overall efficiency.

The remaining chapters are about modelling the boundaries. Chapter 5 presents new

methods for constructing and analysing formulations of locally reacting surfaces that can

be used in FDTD simulations of acoustic spaces. Novel FDTD formulations of frequency-

independent and simple frequency-dependent impedance boundaries are proposed for 2D

and 3D acoustic systems, including a full treatment of corners and boundary edges. The

proposed boundary formulations are designed for virtual acoustics applications using a

rectilinear, nonstaggered grid, and apply to FDTD as well as Kirchhoff variable digital

waveguide mesh methods. These models include simple frequency-dependent boundaries

in which the wall is characterised by a complex impedance expression that incorporates lin-

ear resistance, inertia, and restoring forces. In addition, a new analytic evaluation method

that accurately predicts the reflectance of numerical boundary formulations is proposed.

The results obtained from numerical experiments and numerical boundary analysis (NBA)

are analysed in time and frequency domains in terms of the pressure wave reflectance for

different angles of incidence and various impedances. The proposed boundary models are

compared with the frequency-independent 1D boundary model commonly applied to ter-

minate the digital waveguide mesh [90] and Botteldooren’s boundary model for staggered

Yee’s grid [15].

The extension to modelling generally frequency-dependent boundary models is pro-

posed in chapter 6. The proposed approach allows direct incorporation of a digital

impedance filter (DIF) in the multidimensional (i.e. 2D or 3D) FDTD boundary model

of a locally reacting surface. An explicit boundary update equation is obtained by care-

fully constructing a suitable recursive formulation. The method is analysed in terms of

pressure wave reflectance for different wall impedance filters and angles of incidence. Its

performance is compared with the performance of the 1D model in which a reflectance

filter is combined with the FDTD room interior implementation using KW-pipes [53, 75].

In Chapter 7, the formulation of a novel digital impedance filter model for any member

of the family of 2D and 3D compact explicit FDTD schemes is proposed. Since the interpo-

lated scheme equation represents the most general form of compact explicit schemes, such

a boundary formulation is in this thesis referred to as the interpolated digital impedance

filter model. Such a formulation naturally encompasses the boundary models for all other

compact explicit schemes, which are obtained by setting the values of the respective free

parameters.

In Chapter 8, a method for modelling diffusive boundaries in finite difference time

domain (FDTD) room acoustics simulations with the use of digital impedance filters is

proposed. The presented technique is based on the concept of phase grating diffusers,


and is suitable for modelling scattering from small irregularities in the boundary surface

and diffusers consisting of narrow wells. A range of diffuser types is investigated through

numerical experiments, generally giving good agreement with theory. It is proposed that

irregular surfaces are modelled by shaping them with Brownian noise, giving good control

over the sound scattering properties of the simulated boundary through two parameters,

namely the spectral density exponent and the maximum well depth.

1.4 Contributions

The main contributions of this thesis are as follows:

• A new compact explicit FDTD scheme is identified - named the interpolated wide-

band scheme - which provides the full bandwidth in 2D and 3D simulations, exhibits

no dispersion error in axial directions, and is shown to be an excellent choice regard-

ing accuracy and efficiency.

• The formulation of the boundary condition in terms of pressure only, which applies

to schemes based on unstaggered grids, is proposed.

• A new frequency-independent boundary model of a locally reacting surface for a

family of compact explicit schemes is proposed.

• Novel formulation of simple frequency-dependent walls incorporating linear resis-

tance, inertia and restoring forces is proposed.

• The digital impedance filter (DIF) boundary model is introduced - a new method

for modelling generally frequency-dependent boundaries of a locally reacting surface

type. A structurally stable and efficient explicit boundary formulation is constructed

by carefully combining the boundary condition in the direction normal to the bound-

ary surface with the compact explicit update equation.

• All the boundary models proposed in this thesis include physically-correct formula-

tions of corner and boundary edge nodes, which appears to have never been addressed

in the literature on FDTD/DWM room acoustics simulations.

• Numerical Boundary Analysis (NBA) is proposed - a new analytic method for the

exact prediction of the numerical boundary reflectance of multidimensional boundary

models, such as those proposed in this thesis, thus removing the need for carrying

out elaborate numerical experiments to evaluate the boundary performance.

• A useful method for modelling phase grating diffusive boundaries by designing

boundary impedance filters from normal-incidence reflection filters with added delay


is proposed. These added delays, that correspond to the diffuser well depths, are

varied across the boundary surface, and implemented using Thiran allpass filters.

This technique is suitable for modelling high frequency diffusion caused by small

variations in the surface roughness and, more generally, diffusers characterised by

narrow wells with infinitely thin separators.

In addition, this work also includes the following minor contributions:

• The application of the fourth-order accurate nonstaggered compact implicit scheme

implemented using alternating direction implicit technique is proposed for the first

time in the field of audio and acoustics. This method constitutes an efficient al-

ternative to explicit methods when an extremely high accuracy of the 2D or 3D

simulations is required.

• The most accurate and isotropic compact schemes are identified, and most efficient

ones indicated.

• The most accurate and isotropic in numerical reflectance digital impedance filter

boundary models are identified.

• A method to control sound scattering properties in numerical simulations to match

diffusion coefficient data by shaping surface roughness with a Brownian noise is

proposed.

1.5 Related Publications

Some parts of the work presented in this thesis have been published in the form of con-

ference proceedings and journal articles.

Conference proceedings

1. K. Kowalczyk and M. van Walstijn, “On-line simulation of 2D resonators with re-duced dispersion error using compact implicit finite difference schemes,” Proc. IEEEInt. Conf. on Acoustics, Speech and Signal Process. (ICASSP), pp.285-288, April2007, Honolulu, Hawaii.

2. K. Kowalczyk and M. van Walstijn, “Formulation of a locally reacting wall in finitedifference modelling of acoustic spaces,” Int. Symp. on Room Acoustics (ISRA),pp.1-6, September 2007, Seville, Spain.

3. K. Kowalczyk and M. van Walstijn, “Virtual room acoustics using finite differencemethods. How to model and analyse frequency-dependent boundaries?,” Proc. IEEEInt. Symp. on Communications, Control and Signal Process. (ISCCSP), pp.1504-1509, March 2008, St. Julians, Malta.


4. K. Kowalczyk and M. van Walstijn, “Modeling frequency-dependent boundaries asdigital impedance filters in FDTD and K-DWM room acoustics simulations,” 124thConvention of the Audio Eng. Soc., prepring no. 7430, May 2008, Amsterdam, TheNetherlands. An extended manuscript appeared in the Journal of the AES.

5. M. van Walstijn and K. Kowalczyk, “On the numerical solution of the 2D wave equa-tion with compact FDTD schemes,” Int. Conf. on Digital Audio Effects (DAFx),pp. 205-212, September 2008, Espoo, Finland.

Journal articles

1. K. Kowalczyk and M. van Walstijn, “Modeling frequency-dependent boundaries asdigital impedance filters in FDTD and K-DWM room acoustics simulations,” J.Audio Eng. Soc., vol. 56, No. 7/8, pp. 569-583, July/August 2008.

2. K. Kowalczyk and M. van Walstijn, “Formulation of a locally reacting wall inFDTD/K-DWM modeling of acoustic spaces,” Acta Acustica united with Acustica,accepted for publication in the special issue on Virtual Acoustics, vol. 94, No. 6,pp. 891-906, November/December 2008.

3. K. Kowalczyk and M. van Walstijn, “Wideband and isotropic room acoustics simu-lation using 2D interpolated FDTD schemes,” IEEE Trans. on Audio, Speech andLanguage Processing, accepted for publication.

4. K. Kowalczyk, M. van Walstijn, and D.T. Murphy, “A phase grating approach tomodelling surface diffusion in FDTD room acoustics simulations,” IEEE Trans. onAudio, Speech and Language Processing, submitted for publication.

11

Chapter 2

Fundamentals of Room Acoustics

The aim of this chapter is to briefly review the basics of acoustics related to sound prop-

agation in enclosed spaces. The main focus is on the properties of the medium, leading

to the wave equation, and next on the boundary condition and analytic formulae for

the boundary impedance of a locally reacting surface. The final part reviews currently

available diffusers and the measurement setup for capturing diffusion coefficient.

This chapter is structured as follows. Firstly, the most important acoustic laws appli-

cable to room acoustics are presented in Section 2.1, followed by sound pressure level defi-

nition in Section 2.2. The theoretical formulation of a locally reacting surface is presented

in Section 2.3, including the definition of the boundary impedance and the derivation of

a reflection coefficient. In Section 2.4, an analytic method to calculate modal frequencies

of rectangular room with completely rigid walls is discussed. Section 2.5 provides analytic

formulae for the impedance of acoustic porous materials. An overview of available diffusers

is provided in Section 2.6, focusing on their structure and diffusive properties. Finally, the

setup for diffusion coefficient measurements is discussed in Section 2.7.

2.1 Wave Equation

When a sound wave propagates, the particles of the medium undergo vibrations about

their mean positions. In some regions they may be pushed together, whereas in others

they are pulled apart. Once the wave has passed, the particles return to their original

state. Consequently, the variations of both pressure and velocity occur as functions of time

and space. Sound pressure is defined as a difference between the instantaneous pressure

and the static pressure. The velocity of particle displacement is yet another important

quantity characterising a travelling sound wave.

Even though in large concert halls some variations of temperature cannot be avoided

and air conditioning systems may cause air not to be completely at rest, such inhomo-

Chapter 2. Fundamentals of Room Acoustics 12

geneities are relatively small and can be neglected. Therefore, it seems justified to assume

that the air in the interior of the room can in ideal conditions be regarded as homogeneous

and at rest.

In such a homogeneous isotropic loss-free medium, sound velocity is constant with

reference to time and space. Under these conditions, the magnitude of sound velocity c in

m/s is given as [64]

c = (331.4 + 0.6θ), (2.1)

where θ is the temperature in centigrade. Sound wave propagation in air is governed by

two basic laws, namely the conservation of mass and the conservation of momentum [64].

The former is expressed by

∇p + ρ∂u

∂t= 0, (2.2)

and the latter is given by∂p

∂t+ κ∇u = 0, (2.3)

where p denotes the acoustic pressure, u is the vector particle velocity, ρ is the air density,

c is the sound velocity, and the adiabatic exponent is given by

κ = ρc2. (2.4)

In these equations, we assume that time dependent changes in particle velocity are small

compared to the static values and that particle velocity is substantially smaller than the

sound velocity. These linear equations are typically used to describe practical conditions

for room acoustics. Note that this assumption is typically made in room acoustics, and it

does not hold for high-amplitude sound such as that produced by jet engines. The wave

equation can be derived by eliminating the particle velocity from Equation (2.2) using

Equation (2.3), which yields∂2p

∂t2= c2∇2p, (2.5)

where ∇2p is given as

∇2p =∂2p

∂x2, (2.6)

∇2p =∂2p

∂x2+

∂2p

∂y2, (2.7)

∇2p =∂2p

∂x2+

∂2p

∂y2+

∂2p

∂z2, (2.8)

in a 1D, 2D, and 3D acoustic system, respectively; x, y, and z are directions of an x-

y-z Cartesian coordinate system. This differential equation is fundamental in the field


of acoustics and applies to waves of any type of the wavefront. Furthermore, it holds

not only for sound pressure variations but also for density and temperature variations.

By applying the Fourier transform to the wave equation given by Equation 2.5, the time

invariant version of the wave equation, known as the Helmholtz equation, results

△ p + k2 p = 0, (2.9)

where k denotes the wave number of the wave that is given by

k =ω

c, (2.10)

and ω is the angular frequency. The frequency of vibration is given as f = 2π/ω in Hz,

whereas the wavelength can be calculated from

λ =c

f. (2.11)

The wave equation in theory determines the sound pressure at all positions and at all

times. However, it can only be solved analytically for very special cases for predescribed

boundary conditions. Therefore, numerical techniques are necessary to approximate solu-

tions of the wave equation for more general acoustic spaces.

2.2 Sound Pressure Level

As has been mentioned in Section 2.1, sound pressure is a difference between the pressure

caused by the passing wave and the ambient pressure at a particular point in space. The

effective sound pressure is the root mean square (RMS) of such a pressure difference

measured over a period of time at the point in space, according to

prms =

√p1

2 + p22 + ... + pn

2

N, (2.12)

where p1, p2, ..., pn is the instanteous pressure in Pa measured over N samples. Due to

the nature of the human hearing, sound pressure is often expressed on a logarithmic scale

in relation to a reference pressure value po by

L = 20log∣∣∣prms

po

∣∣∣, (2.13)

where L denotes the level difference between two sound pressure values, measured in dB.

If the reference pressure value is taken as po = 2 10−5N/m2, which is the threshold of

hearing at a frequency of 1kHz, the resulting value L is the ‘sound pressure level’.


2.3 Locally Reacting Surfaces

In general, room acoustics concerns sound propagation in enclosures, where a medium is

bounded by side walls, floor and ceiling. Most boundaries in rooms reflect some portion of

the impinging energy, and some fraction of energy is absorbed. In this section, we consider

a reflection of a plane sound wave from a single surface.

Concerning the shape of the incident wave, we assume an incident wave to be plane,

that is the wave is propagating in one direction only. The name - plane wave - stems from

a planar surface of a constant phase which is perpendicular to the propagation direction.

Plane waves are actually hard to encounter in reality. In real room acoustics, we primarily

deal with spherical waves or sections of spherical waves [64]. The most similar to plane

waves is the wavefield caused by an infinite number of monopoles distributed along a line,

for which a cylindrical wave results [64]. However, when a reflecting wall is sufficiently

far away from the source position, the curvature of the wavefront can be neglected and

the resulting error caused by substituting a spherical wavefront with a plane wave can be

considered negligible.

The wall considered in this section is assumed to be unbounded and plane. However,

very small irregularities (i.e., roughness of the surface that is much smaller than the wave-

length of the incident sound wave) along a wall that is much larger then the wavelength

are not excluded in this consideration.

A wave reflected from a wall has both phase and amplitude that differ from those of

an incident wave. The incident and reflected waves interfere with each other creating (at

least partially) a standing wave. We can express such changes with a reflection coefficient

R defined as a function of frequency, herein also referred to as reflectance in order to

emphasize its frequency-dependency. Such a reflectance completely defines the acoustic

properties of a wall for any frequency and angle of incidence.

2.3.1 Boundary Impedance

Two prime quantities in room acoustics theory are sound pressure and particle velocity.

The first one is a scalar, whereas the second is a vector quantity. For convenience, the scalar

value defined as the normal component of the particle velocity has been introduced. The

term normal refers to both wavefront or the boundary surface when a sound wave encoun-

ters a wall. At a right boundary (depicted in Figure 2.1), the ratio between the pressure

and the normal component of the particle velocity defines the boundary impedance

Zw =p

ux, (2.14)


where p denotes pressure and ux is the velocity component that is normal to the surface

of the boundary. The boundary impedance is generally complex as the reflection alters

both amplitude and phase of an incident wave. The boundary impedance is often divided

by the characteristic impedance of air

ξw =Zw

ρc, (2.15)

in which case it is referred to as the specific acoustic impedance. The typical value for the

characteristic impedance of air at normal condition is [64]

ρoc = 414kgm−2s−1. (2.16)

The inverse of the specific acoustic impedance is the specific acoustic admittance defined

as

Yw =1

ξw. (2.17)

The intensity of the reflected wave is reduced by |R|2 in comparison to the incident

wave. Based on this property, an alternative quantity to the reflection coefficient can be

formulated. This quantity is referred to as the absorption coefficient and is given as

α = 1 − |R|2. (2.18)

Note that the absorption coefficient only defines the change in amplitude but does not

include any information about the phase change.

2.3.2 Reflection at Normal Incidence

In this section, we explore the relationship between the boundary impedance and reflection

coefficient at normal incidence, largely following the derivation in [64]. Let us first consider

a wall parallel to the x-axis of a rectangular coordinate system x-y. An incident plane wave

is travelling in the positive x-direction, as depicted in Figure 2.1, for which the pressure

and velocity component normal to the wall are given respectively as

pi(x, t) = Po ej(ωt−kx) (2.19)

and

ui(x, t) =Po

ρcej(ωt−kx). (2.20)

When such an incident wave interacts with the boundary at normal incidence, the

propagation direction is reversed. In addition, the amplitude of the reflected wave de-

creases due to boundary absorption and phase undergoes a change. In practice, phase


x=0

pi

pr

Figure 2.1: Plane wave reflection for a right boundary located at x = 0 at normal angle of incidence.

alterations occur for nonrigid walls such as curtains, light nonstiff walls, wall and floor

coverings. Both these changes are fully defined by the reflection coefficient R. Further-

more, the flow is reversed, and hence a change in sign of the particle velocity is required.

Thus, the pressure and particle velocity of the reflected wave are given as

pr(x, t) = R Po ej(ωt+kx) (2.21)

ur(x, t) = −RPo

ρcej(ωt+kx), (2.22)

respectively. The total sound pressure and particle velocity in front of the wall are obtained

by adding the respective values of the incident and reflected waves. In addition, we assume

that the boundary is located at x = 0 for simplicity. Consequently, the total pressure and

particle velocity in the plane of the wall is

p(0, t) = (1 + R) Po ejωt (2.23)

u(0, t) = (1 − R)Po

ρcejωt. (2.24)

Dividing the pressure value by the normal component of the particle velocity u yields the

boundary impedance

Zw = ρc1 + R

1 − R, (2.25)

from which the reflection coefficient R can be found as

R =Zw − ρc

Zw + ρc=

ξw − 1

ξw + 1. (2.26)

Three extreme cases for the value of the boundary impedance are:


x=0

pi

pr

Figure 2.2: Plane wave reflection for a boundary located at x = 0 at oblique angle of incidence.

• A hard wall, in which case the boundary is infinitely rigid (|Zw| → ∞). Thus the

reflection coefficient amounts to R = 1 and total reflection occurs.

• A soft wall is characterised by Zw = 0, in the case of which R = −1. This time,

reflection is also total but out of phase.

• When the boundary impedance equals the characteristic impedance of the medium

Zw = ρc, R = 0 and a completely absorbent wall is obtained.

2.3.3 Reflection at Oblique Incidence

Let us consider an incident wave propagating in the direction x′, for which the deviation

from the direction normal to the wall is given by an angle θ, as illustrated in Figure 2.2.

Again, without the loss of generality this problem can be treated in two dimensions. Such

a propagation direction is related to the x-y coordinate system by

x′ = (x cos θ + y sin θ). (2.27)

The pressure pi and the particle velocity component ui of an incident wave that is normal

to the boundary are given respectively by

pi(x, y, t) = Po ejωt e−jk(x cos θ+y sin θ) (2.28)

ui(x, y, t) =Po

ρccos θ ejωt e−jk(x cos θ+y sin θ). (2.29)

Similarly to the case of normal incidence reflection, the sign of x is reversed in the

reflected wave because of the change of the propagation direction, and amplitudes are

amended respectively. Consequently, the pressure and normal velocity component of the


reflected wave are

pr(x, y, t) = R Po ejωt e−jk(−x cos θ+y sin θ), (2.30)

ur(x, y, t) = −RPo

ρccos θ ejωt ejk(−x cos θ+y sin θ). (2.31)

By setting x = 0 at the wall, and adding the pressure and velocity components of both

incident and reflected waves, the total values along the boundary surface are obtained

p(0, y, t) = (1 + R) Po ejωt e−jky sin θ, (2.32)

u(0, y, t) = (1 − R)Po

ρccos θ ejωt e−jky sin θ (2.33)

Finally, the boundary impedance is obtained by dividing the total pressure by the total

normal velocity component

Zw =ρc

cos θ

1 + R

1 − R, (2.34)

from which the reflection coefficient is obtained as

R =Zw cos θ − ρc

Zw cos θ + ρc, (2.35)

or, expressed in terms of the specific boundary impedance, it is given as

Rθ =ξw cos θ − 1

ξw cos θ + 1. (2.36)

2.3.4 Discussion

Most of the boundaries in enclosed spaces are solids, such as concrete brick walls, in

which case additional shear waves are excited for an oblique-incident sound wave [82].

Furthermore, there are several types of waves travelling on the boundary surface, Rayleigh

waves are good examples. Several types of transverse wave motions of the solid should be

taken into account when the width of the solid is much smaller than boundary dimensions

[8]. Consequently, the reflection coefficient would have to be replaced with a reflection

function that defines the reflected wave at one position when the boundary is excited at

another position. The theoretical treatment of a vibrating panel on a wall in a rectangular

room is provided in [82]. However, such nonlocally reacting walls are hard to treat properly

in practice due to the lack of detailed measurement data and modelling problems.

In the context of room acoustics, the boundary impedance is often assumed to be in-

dependent on the angle of the incident sound wave. This simplification is only true for

walls in which the particle velocity at the boundary surface depends solely on the sound

pressure in front of the wall element, and not on the pressure of neighbouring elements

[64]. Such walls are referred to as locally reacting surfaces. The locally reacting surface is


encountered when a wall itself and the space behind the wall does not allow wave prop-

agation in the direction parallel to the boundary surface; seat and floor coverings, heavy

curtains, and light nonstiff walls are good examples. In the context of computer simula-

tions of room acoustics, the assumption of local reaction reduces the diffusive properties of

simulated walls, decreasing the overall diffusiveness of the simulated space. Consequently,

techniques for modelling diffusion can be used to compensate for the lack of real nonlocally

reacting boundaries.

2.4 Eigenmode Model (for Rigid Walls)

This section deals with searching for the solutions of the wave equation using series of

eigenmodes and eigenfunctions. Even though eigenmodes occur for enclosures of arbitrary

shapes, analytic solutions can only be found for special cases of the room geometry and

simple boundary impedance values. For instance, complex eigenvalues result for a complex

boundary impedance, and the solution cannot generally be analytically found without

further approximations.

For a rigid boundary (ξw → ∞), the normal velocity component is zero, and thus the

boundary condition reduces to∂p

∂n= 0. (2.37)

Consider an acoustic space of a rectangular geometry and dimensions (Lx, Ly, Lz), in

which all walls are parallel to the axis of the Cartesian coordinate system. The Helmholtz

equation can be solved by separation of variables and composing the solution of three

factors

p(x, y, z, ω) = px(x, ω)py(y, ω)pz(z, ω), (2.38)

each depending solely on one propagation direction. The Helmholtz equation is split into

three ordinary equations. For instance px satisfies equation

∂px

∂x+ k2

xpx = 0, (2.39)

where kx denotes the wavenumber in x-direction. Furthermore, the separation of variables

applies to the boundary condition, i.e.

∂px

∂x= 0 (2.40)

for x = 0 and x = Lx. Analogous conditions apply to y- and z-directions, and the

respective directional wavenumbers are related to each other by

k2 = k2x + k2

y + k2z . (2.41)


The solution that satisfies the boundary condition given by Equation (2.40) is px =

cos(kxx), in which the allowed values for the wavenumber kx are

k2x =

nxπ

Lx, (2.42)

where nx is a nonnegative integer. Consequently, the eigenvalues of the wave equation are

given by

knxnynz = π[(nxπ

Lx

)2+(nyπ

Ly

)2+(nzπ

Lz

)2] 1

2

, (2.43)

and associated eigenfunctions are given as

pnxnynz(x, y, z) = cos(nxπx

Lx

)+ cos(

nyπy

Ly

)+ cos(

nzπ

Lz

). (2.44)

Equation (2.44) multiplied by ejωt represents a 3D standing wave. Standing waves are

referred to as room modes and their respective frequencies as modal frequencies. The

eigenfrequencies corresponding to the eigenvalues are given as

fnxnynz =c

2πknxnynz . (2.45)

2.5 Acoustical Porous Material

A common practice in room acoustics is taming unwanted reflections in an enclosed space.

For that purpose an absorptive material is often applied to walls and other reflective

surfaces. The most popular choices include dense porous materials such as polyurethane

foam and fiberglass. Carpet and drapes are examples of soft fibrous materials, which are

commonly applied as room fittings in order to absorb high frequencies. The sound is

absorbed in porous material by converting the acoustical energy into small amounts of

heat. In order to simulate such materials in computer simulators, information about the

boundary impedance of porous materials is necessary.

Delany and Bazley [29] have proposed empirical expressions for characteristic impedance

Zw(ω) and the propagation constant Γ(ω) for porous materials. Their formulation is based

on a large number of measurements of fibrous materials with porosities. The porosity is

defined as the ratio of the fluid volume occupied by the continuous fluid phase to the total

volume of the porous material.

If we denote the static air flow resistivity as σ expressed in Nm−4s, the expressions

proposed by Delany and Bazley for the impedance and the propagation constant at normal


room conditions are as follows

Zw(ω) = ρc[1 + 0.0571

(ρf

σ

)−0.754

− j0.087

(ρf

σ

)−0.732 ], (2.46)

Γ(ω) =jω

c

[1 + 0.0978

(ρf

σ

)−0.70

− j0.189

(ρf

σ

)−0.595 ]. (2.47)

According to [29], such boundary expressions are valid in the following range of the

flow resistivity values

0.01 <f

σ< 1.00, (2.48)

however, some discrepancy with measurements has been observed at low frequencies. In

general however, Equations (2.47) and (2.48) provide quite accurate estimations and are

widely used due to the simplicity. Note that the acoustic behavior of the fibrous material

is described by one parameter only, namely the flow resistivity σ.

Novel theoretical formulations for viscous forces in porous media has been proposed in

[47]. This enabled a new formulation of the characteristic impedance and the propagation

constant for a particular geometry of such porous materials. The equations proposed in [5]

are more physical as they are based on a physical representation of acoustical phenomena

that take place in porous materials. Consequently, a significant improvement in accuracy

at low frequencies has been obtained, where the equations proposed by Delany and Bazley

give unphysical predictions.

Taking into account the inertia and the viscous forces per unit volume of the air in the

material, the equation for the dynamic density ρ(ω), expressed in kg/m3, can be written

as [5]

ρ(ω) = 1.2 +[− 0.0364(

ρf

σ)−2 − j0.01144(

ρf

σ)−1] 1

2

. (2.49)

The relation of the divergence of the averaged molecular displacement of air and the

averaged variation of the pressure is called the dynamic bulk modulus, which at normal

room conditions is given as

K(ω) = 101320j29.64 +

[2.82(ρf

σ )−2 + j24.9(ρfσ )−1

]1

2

j21.17 +[2.82(ρf

σ )−2 + j24.9(ρfσ )−1

]1

2

, (2.50)

expressed in N/m2. Equations (2.49) and (2.50) next serve as a basis to obtain analytic

formulae for the characteristic impedance Zw(ω) and propagation constant Γ(ω) of the

porous material, which are expressed as

Zw(ω) =[ρ(ω)K(ω)

] 1

2

(2.51)


and

Γ(ω) = j2πf[ ρ(ω)

K(ω)

] 1

2

(2.52)

Finally, the specific wall impedance ξw(ω) of a rigidly backed layer can be calculated from

ξw(ω) =Zw(ω)

ρccoth [Γ(ω)d] , (2.53)

where d is the layer thickness. Due to the fact that both Γ and Zw depend on the flow

resistivity, the boundary impedance is fully characterised by the two parameters d and σ.

Similarly to the empirical formulations of Delay and Bazley, these equations are only valid

under assumption that the porosity is smaller than unity.

2.6 Diffusers

Sound scattering is a fundamental phenomenon encountered in room acoustics that plays

a crucial role in defining the acoustic properties of spaces. It encompasses edge diffraction

effects as well as diffusive reflection of sound waves from not perfectly smooth/uniform sur-

faces. Diffusion of sound in rooms is perceptually important because it reduces colouration

and interaural coherence, resulting in subjective listener preference [99, 21]. Application

of sound diffusers can greatly help improving the acoustic communication properties of

spaces, for example clarity of speech, or sense of ensemble in music performance [25].

Consequently, a broad number of diffusive surfaces has been developed over the past

40 years where research efforts have been concentrated on developing the most effective

diffusers. The main goal of designing a diffuser is to reduce the energy reflected in the

specular reflection and ensure spatial scattering in many directions rather than just redi-

recting the sound in one direction that is different from the specular direction. As a

measure of effectiveness, the diffusion coefficient is used. The higher the value of the dif-

fusion coefficient, the better the diffusive surface structure is. In this section, an overview

of geometrical diffusers is presented.

2.6.1 Maximum Length Sequence

Early diffusers are based on pseudorandom noise theory, in particular periodic sequences

characterised by a flat power spectrum. The first diffuser designed with the aim to scatter

an incident plane wave evenly is the maximum length sequence (MLS) diffuser proposed

by Schroeder [98], in which the local normal-incidence reflection coefficient values change

according to the pseudorandom sequence between −1 and 1. An example maximum length


λo

4

Figure 2.3: Lateral cross section through a maximum length sequence of length 13.

sequence of length 13 is illustrated in Figure 2.3, which reads

(+1,+1,+1,+1,+1,−1,−1,+1,+1,−1,+1,−1, +1). (2.54)

Neglecting diffraction effects, the local reflection coefficient −1 is in practice realised by

creating wells that are one-quarter design wavelength deep in a totally rigid wall. There-

fore, the diffusive properties of an MLS diffuser depend on the wavelength of an incident

sound [98]. A relatively good diffusion is only obtained for half an octave below and above

the design frequency. For instance, near-specular reflections are apparent for half the de-

sign frequency, for which wells are half a wavelength deep giving an effective reflection

that equals 1 [98].

2.6.2 Quadratic Residue Diffuser

Further research by Schroeder into diffusers that are characterised by an increased range

of frequencies where sound scattering is still apparent has led to M -ary maximum length

sequences [99]. These sequences imply that wells of several different depths should be

applied, and good diffusion across wide frequency ranges should result. Quadratic residue

sequences are given as [99]

sn = n2modN, (2.55)

where N is a prime number and n2modN is the least nonnegative remainder of modulo

N . Such sequences are periodic with period N and symmetric around [n = 0 and n =

(N − 1)/2] [99]. For instance, the quadratic residue sequence for N = 17 reads

(0, 1, 4, 9, 16, 8, 2, 15, 13, 13, 15, 2, 8, 16, 9, 4, 1, ...). (2.56)

One of the great properties is that the Fourier transform rn of the locally changing reflec-

tion coefficient

rn = exp j2πsn

N, (2.57)

has a constant magnitude, i.e.

|rn| =1

N. (2.58)


dn

wNw

Figure 2.4: Lateral cross section through a quadratic residue diffuser with N = 7. The well width anddepths are denoted respectively as w and d.

Consequently, in theory the energy is spread evenly in all directions making the QRD an

optimum diffuser. However, in practice this is not the case. The quadratic residue surface

is a good scatterer for frequencies higher than the design frequency, in particular at integer

multiplies of the design frequency up to a limit given by prime number N . However, for

many intermediate frequencies the QRD behaves similarly to a flat surface [24]. A more

stringent limit at high frequencies is given by the well width w, namely [99]

λo > 2w. (2.59)

On the other hand, good scattering should not be expected for frequencies even half an

octave below the design frequency [99]. The depths of the quadratic residue diffuser are

given by

dn =λo

2

sn

N, (2.60)

where λo is the design wavelength. An example quadratic residue diffuser for the sequence

of length 7 is depicted in Figure 2.4. The majority of commercially available QRDs are 1D,

that is their well depths are changing in one direction only (see Figure 2.5). Such diffusers

are mainly used for lateral scattering. Thin rigid separators between individual wells are

very important in the scattering process, especially for oblique incidences [99]. The lack

of these fins would greatly diminish scattering properties resulting in poor diffusion.

Alternatively, QRD could be constructed as a reflecting planar hard wall with vary-

ing local impedance in a periodic fashion, where the reflection coefficients are given by

Equation (2.57). However, such rapid changes in the boundary impedance along the wall

surface cannot be manufactured [99].

The 1D quadratic residue diffuser can be extended to 2D by replacing a quadratic

residue sequence sn in Equation (2.60) with two quadratic residue sequences sk and sl,


Figure 2.5: 1D quadratic residue diffuser.

which yields

dn =λo

2

sk + sl

N, (2.61)

where sl = l2 and sk = k2, l = 0, 1, 2, ... and k = 0, 1, 2, ....

2.6.3 Modulated Quadratic Residue Diffuser

Quadratic residue sequences are of finite length, and so is the size of a standard manufac-

tured quadratic residue diffuser. Therefore, in order to cover a large wall area, individual

diffusers are typically concatenated. However, repetition of the sequence causes a more

concentrated reflection of energy into gratings, the sharpness of which depends on the

number of repeats. The more repeats, the sharper the lobes. A solution similar to mod-

ulating the carrier in communication systems has been proposed by Angus [7], which is

based on using a pseudorandom spreading sequence.

Modulated Phase Reflection Grating

The modulated phase reflection grating is a modulation technique which is based on using

a normal and inverted version of the basic sequence according to the pseudorandom binary

sequence, such as MLS or Barkley codes [7]. As a result, the diffusion comparable to the

scattering of a single diffuser is achieved. Such an inverted diffuser can be viewed as an


upside down version of the original diffuser. Furthermore, an inverted sequence is modulo

N equivalent to the normal sequence if zero depths in the inverted diffuser are replaced

with maximum depths [7]. For instance, the original quadratic residue sequence of length

5 is

(0, 4, 1, 1, 4), (2.62)

and its inverted version is

(5, 1, 4, 4, 1). (2.63)

Sequence inversion modulated gratings greatly reduce the effect of narrow energy lobes

caused by periodicity of the sequence. However, the diffusion characteristic is not enhanced

compared to having just one quadratic residue diffuser, and hence high scattering can only

be expected at integer multiplies of the design frequency.

Orthogonal Modulation

Orthogonal modulation is based on interconnecting two quadratic residue diffusers of dif-

ferent lengths but with the same maximum depth. This way a variation in the step

size is introduced [7]. Similarly to the modulated phase reflection grating technique, the

maximum length sequence is typically applied as a pseudorandom binary sequence.

The well depths of the orthogonally modulated quadratic residue diffuser are given by

dn = n2modN(dmax

N) n = 0, 1, 2, ..., N (2.64)

where N is a sequence length and dmax is the well depth that corresponds to a sequence

value equal to N . For instance, an orthogonal modulation of sequences of lengths 5 and 7

reads

dn =(0,

1

5,4

5,4

5,1

5, 0,

1

7,4

7,2

7,2

7,4

7,1

7

)dmax. (2.65)

Such a composite structure, which results from orthogonal modulation, generally gives

better diffusion for higher frequencies since there are now two integer multiplications of

the design frequency [7]. In fact, it does not really behave like a flat panel up to the

frequency determined by the lowest common multiple of the sequence lengths. Such a flat

panel frequency is given by [7]

fflatplate =N1 N2 c

2dmax, (2.66)

where N1 and N2 are the two respective quadratic residue sequence lengths. A more

stringent limit at high frequencies may come from the widths of the diffuser wells.


Figure 2.6: Lateral cross section through a diffractal based on a quadratic residue sequence N = 7.

2.6.4 Diffractals

Addressing the issue of diffusion for bass and high frequencies simultaneously is possible

with the use of diffractals (i.e., diffusing fractals). Diffractals are constructed by embedding

small scaled versions of the quadratic residue diffuser at the bottom of each well of a larger

quadratic residue diffuser [26]. An example diffractal is illustrated in Figure 2.6. A small

scaled version of the diffuser scatters mid-high frequencies, whereas a larger diffuser deals

with the scattering of the bass frequencies. Both diffusers are orthogonal and independent

in performance as large wavelengths are unaware of the small surface irregularities of the

high frequency diffuser, and the polar distribution for short wavelengths remains unaffected

by the phase changes introduced by the low frequency diffuser. Diffractals provide an easy

means of introducing an additional low-frequency absorption along with low frequency-

diffusion by fitting damped diaphragmatic membranes or thick porous layers to the rear

of the diffractal [26].

2.6.5 Curved Diffusers

Schroeder diffusers scatter sound uniformly for oblique angles of incidence. However,

strong equalising flows develop between the elements of the quadratic residue diffuser

causing an additional absorption at low frequencies [40], [65]. Such extra bass absorption

is usually advantageous in studio spaces, but in large concert halls additional absorption

may actually be disadvantageous. Therefore, simple curved reflectors based on the arc

of a circle have been suggested as an alternative, and the optimisation techniques have

been proposed in order to find the best diffusers for covering large wall areas [20]. It

is important to reduce the number of curvatures in the design of such surfaces so that

additional absorption is not introduced.

Through optimisation, a rough semicircular shape has been found the best curved

diffuser if the width of a diffuser does not exceed 1m [20]. Such a simple semicircular

shape produces fairly uniform diffusion for on-axis sources but scattering is less uniform

for oblique incidences. Furthermore, concatenating diffusers that are semicircular in shape


again reduces their diffusive properties. Therefore, for wider diffusers (e.g. diffusers which

are 4m wide), the optimised curvature structures prove better scatterers than a simple

semicircular shape [20].

2.6.6 Fractional Brownian Diffusers

Good sound scatterers can be realised with the use of the Fourier synthesis technique, in

which a Gaussian white noise signal is spectrally shaped with the use of a linear roll off

filter [22]. The decrease in the spectral content of the input signal depends on the gain

of the filter at each frequency band. Since the shape generated by the Fourier synthesis

technique represents Brownian motion, such sound scatterers are referred to as fractional

Brownian diffusers. A simple roll off filter is given as

A(f) =1

fβ/2, (2.67)

where β is the spectral density exponent. For β = 1, pink noise results in which the

roll off is of 3dB/octave. For higher values of the spectral density component, Brownian

noise is obtained in which the sharpness of spikes decreases with β. For instance, a roll

off of 6dB/octave is achieved for β = 2. In order to obtain the best performance at low

frequencies, the spectral density component should be large, leading to a smoother shape.

Good high frequency scattering results for a more spiky shape which is obtained for low

values of β.

The results presented in [22] indicate that the sound scattering properties of fractional

Brownian diffusers at high and low frequencies are determined to a great extent by the

roll off filter. However, the diffusive quality does not solely depend on the spectral density

component but also on the white noise sequence. Therefore, the optimisation technique has

been proposed in order to have a total control of the diffusive properties. Such optimisation

is also useful for manufacturing purposes, in which a spiky shape is hard to produce.

However, in practice a number of fractional Brownian diffusers have to be repeated to

cover large wall areas, inevitably leading to the decrease in the diffusive properties caused

by the pattern repetition, in a similar fashion to concatenated Schroeder diffusers. By

analogy to fins of Schroeder’s diffusers, the flow of sound energy around sharp spikes may

increase absorption of fractional Brownian diffusers [22].

2.7 Diffusion Coefficient

The diffusion coefficient is a quantity developed in order to evaluate the quality of the

diffuse reflections from a scattering surface. It is a measure of the uniformity of the polar

response of sound scattering across the range of reflection directions. All mechanisms of


Figure 2.7: Setup for the measurement of the diffusion coefficient according to AES standard [3].Squares indicate a changing position of a source and circles indicate receivers.

the diffuse reflection are simultaneously evaluated, including scattering effects caused by

the roughness of the boundary surface and the edge diffraction caused by finite-size of the

boundary element.

The definition and guidelines concerning the measurement of the diffusion coefficient

are provided in the Audio Engineering Society information document for room acoustics

and sound reinforcement systems entitled “Characterisation and measurement of surface

scattering uniformity” [3]. Polar response information, that is the distribution of reflected

energy across all possible reflection directions for a given angle of incidence, is necessary

to compute the diffusion coefficient value. According to [3], scattering properties are

generally evaluated on a single 2D plane of reflection or across the hemisphere depending

on the diffuser. For a single plane diffuser, the measurement should be undertaken in the

plane of maximum diffusion. The source positions and receivers’ positions are placed on

a semicircle in the front plane of the diffuser. In order to measure the random-incidence

diffusion coefficient, the maximum angular resolution of receivers should not exceed 5o,

covering a semicircle around the reference normal. On the other hand, source positions

should be measured with a maximum resolution of 10o on a semicircle with a larger

radius, as illustrated in Figure 2.7. The measurements should ideally be undertaken in

an anechoic chamber to exclude reflections from other surfaces such as walls, floor and

ceilings. Alternatively, a room sufficiently large compared with the object under test can

be chosen as long as an appropriate window is applied to the measured impulse response

so that unwanted side-wall reflections are removed from the measured signal.

One obvious improvement to the measurement procedure presented in [3] would be to

undertake measurements with and without the test sample and the application of a longer

window that also allows first-order reflections from surrounding boundaries. Subtracting

these two impulse responses would remove first-order reflections from side walls. This way

longer input signals characterised by reach spectral content could be applied.

Concerning the test sample, the entire diffuser structure applied in a real acoustic

space should be tested in order to ensure that all the surface roughness diffraction and


edge effects are characterised. However, in the vast majority of cases the whole diffuser

is too large to be fitted in an anechoic chamber. Therefore, according to [3] the size of

the diffuser sample should be large enough so that at least 4 complete repete sequences

of a periodic test surface are included in a sample. For nonperiodic diffusers, the size of

the sample should be sufficient so that surface scattering effects are more prominent than

edge diffraction effects.

Ideally, diffusion coefficient measurements should be undertaken under true far field

conditions for on-axis scattering, which are met when [3]

r ≫ Dmax,r

Dmax≫ Dmax

λ, (2.68)

where Dmax is the largest dimension of the diffuser, λo is the wavelength, r1 is the distance

from the source to the reference point, r2 is the distance from the receiver to the reference

point, and

r =2r1r2

r1 + r2. (2.69)

True far field conditions require large measuring distances, often much larger than can

be realistically achieved. Fortunately, true far field conditions are not necessary. If true

far field conditions cannot be achieved, the general requirement is that at least 80% of

receivers are placed outside of the specular zone, which is defined as the region over which

a geometric reflection occurs [23].

Directional diffusion coefficients for a particular source position are calculated from

impulse responses measured at all receiver positions. The impulse response with the test

surface present h1(t) and without the test surface h2(t) are obtained in two successive

measurements. This way the unwanted background reflections can be removed by sub-

tracting h1(t) from h2(t). Furthermore, an impulse response h3(t) from loudspeaker (used

as a source) to microphone (used as a receiver) should be measured, and next deconvolved

from the subtracted impulse response as

h4(t) = IFT[FT [h1(t) − h2(t)]

FT [h3(t)]

], (2.70)

where FT denotes a forward Fourier transform and IFT denotes an inverse Fourier trans-

form. A rectangular window is next applied, the size of which is determined such that the

full test response is obtained and side-wall reflections are excluded. A windowed impulse

response should be Fourier-transformed and RMS pressure amplitude levels are calculated

in each one-third octave band. Such frequency ranges should conform to the ISO 266

standard [1]. Sound pressure levels, defined as power in each frequency band, serve as a


basis for calculating the directional diffusion coefficient. For a fixed source position and

sound pressure levels Li from n receivers, the directional diffusion coefficient is given as

[3]

dθ =

(n∑

i=1

10Li/10

)2

−n∑

i=1

(10Li/10

)2

(n − 1)n∑

i=1

(10Li/10

)2. (2.71)

Finally, the diffusion coefficient is calculated as an arithmetic average of all directional

diffusion coefficients. If the measurement criteria concerning angular resolutions of source

and receivers’ positions are met, then the random-incidence diffusion coefficient dri is given

as the arithmetic mean of directional diffusion coefficients [3]

dri =

∑180o

i=0o,i=i+10o dθ

19. (2.72)

2.7.1 Discussion

One of the consequences of the diffusion coefficient measurement technique is that large

diffusion coefficient values are obtained at low frequencies even for a completely flat panel.

This is due to edge diffraction of the finite size of the panel. It is therefore a good practice,

when presenting the diffusion coefficient data, to provide the diffusion coefficient values

for a flat plane surface of the same dimensions measured under the same test conditions

as a reference [23].

The finite width of the test sample additionally imposes a constraint on the lowest

frequency that is effectively reflected from the sample. For incident waves that are sub-

stantially longer than the width of the diffuser sample, almost no reflection of energy is

observed and the edge diffraction starts to dominate. Therefore, for a given sample width

one can calculate the lowest cut-off frequency below which the diffusion coefficient data

should not be regarded valid [23].

As a measure of correlation between sound waves scattered in different directions,

the diffusion coefficient is a useful measure of spatial diffusion. However, it does not

monitor temporal dispersion properties. The amount of the sound dispersed in time in the

impulse response of the reflected sound is very important in the sound scattering process.

In particular, time-spreading property of the diffuser is important in the suppression of

flutter echo [21].


2.7.2 Scattering Coefficient

Apart from the diffusion coefficient, there exists a second quantity that deals with scat-

tering properties, namely the scattering coefficient [124, 2]. It has been introduced for

geometrical room acoustic models since the diffusion coefficient cannot be directly applied

to a simplified geometrical approach. The scattering coefficient differentiates between the

portion of energy that is reflected specularly and the energy that is reflected in a diffuse

way. It is defined as the ratio between the scattered energy and the total reflected sound

energy. The scattered energy is usually distributed according to the Lambert’s law, in

which the diffuse sound energy is scattered in a random direction.

As the scattering coefficient contains only partial information about the spatial distri-

bution of the scattered sound, it does not differentiate between dispersion and redirection.

Consequently, high scattering coefficient values can be obtained for a slightly tilted flat

surface. When treating an echo problem, even surfaces with high scattering values may

simply redirect echoes from one place to another, instead of dispersing it. Since the scat-

tering coefficient is only concerned with how much of the reflected energy is moved from

specular directions, it is not considered a good measure of the quality of the diffusive prop-

erties. Consequently, it cannot be used as a quality measure in the design and evaluation

of real diffusers [21].

Furthermore, the wave-based room acoustic simulators are incompatible with the for-

mulation of the scattering coefficient. This is due to the wavelengths that are not physically

consistent with the incoherent energy approach, and consequently the diffusion coefficient

is the only measure of diffusion that can be applied in wave-based room acoustic models,

such as finite difference time domain simulations of sound propagation in acoustic spaces.

2.8 Summary

In this chapter, the basics of room acoustics were briefly reviewed, with the main focus

on fundamental acoustic laws describing sound wave propagation in air and reflections

from boundaries. It is well understood how to define the boundary impedance and the

reflection coefficient for locally reacting surfaces, and therefore only such boundaries are

considered for simulations of room acoustics in this thesis. In addition, analytic formulae

for the impedance of porous materials were discussed, which are used for the evaluation

of boundary models proposed in chapters to follow. Finally, an extensive overview of

commercially available diffusers was presented and the standardised technique to measure

their diffusive properties was described. The remaining question is if we can simulate those.

One of the questions addressed in subsequent chapters is if we can simulate diffusive and

nondiffusive walls.

33

Chapter 3

Elements of Numerical Modelling

There exist numerous numerical techniques that can be applied to modelling acoustic

spaces. Although these techniques have been commonly applied in the context of room

acoustics simulations, research efforts seem to be independent of each other. This is partic-

ularly true for the finite difference time domain technique (FDTD), which can be divided

into subclasses based on rectilinear staggered and unstaggered girds or implemented as

a digital waveguide mesh (DWM). These methods are usually treated separately in the

literature and may sometimes send a confusing message to an inexperienced reader that

these constitute totally distinct techniques. The main purpose of this chapter is to show

that these are not distinct approaches but rather they fall into one larger category of finite

difference time domain methods. This chapter provides an overview of various numerical

modelling techniques and highlights the equivalence of various approaches.

The chapter is structured as follows. An overview of computer-based methods appli-

cable to simulations of room acoustics is provided in Section 3.1, including the motivation

for the chosen technique. Section 3.2 presents the basics of the finite difference time do-

main method. In particular, a basic approximation to the wave equation by means of a

standard leapfrog scheme is presented, followed by stability and dispersion error analysis.

A second type of FDTD schemes based on a staggered Yee’s grid is presented in Section

3.2.3, addressing the issues of stability and dispersion. In addition, Botteldooren’s simple

frequency-dependent boundary model is discussed. In Section 3.2.4, another subclass of

the finite difference method is described, and the equivalence between the digital waveg-

uide mesh and the FDTD implementation is shown. In addition, a literature review of

boundary models available in the DWM literature is provided. The issue of implicit finite

difference methods and their computationally efficient implementation using the alter-

nating direction implicit technique is presented in Section 3.2.5. A useful technique for

reducing the directionally-independent dispersion error is discussed in Section 3.3, followed

by the description of an efficient algorithm for 1D matrix inversion presented in Section

Chapter 3. Elements of Numerical Modelling 34

3.4. Finally, the issue of fractional delay filters with maximally flat magnitude response

that can be applied for modelling phase grating diffusers is discussed in Section 3.5.

3.1 Room Acoustics Modelling Methods

In this section, an overview of room acoustic modelling techniques and a motivation for

the choice of a one particular method for further investigation in this thesis is presented.

There are two main-existing approaches concerning room acoustics modelling, namely,

the physical and the perceptual approach. The former is based on a precise simulation of

physical phenomena in a sound scene based on physical parameters. The latter imitates

rather than accurately simulates an acoustic environment aiming at creating the percep-

tual audible impression of a listener. Typically, with a physical approach, an enclosed

acoustic space is simulated according to the room geometry, reflectivity of boundaries and

directivity of a sound source. The idea is to obtain an impulse response by modelling

sound propagation and reflections in an acoustic space. This approach was originally ap-

plied to architectural acoustic design and acoustic space evaluation. However, it can also

be applied to auralisation by inputting a monophonic, anechoic signal and taking into

consideration the directionality of a sound source and a listener before reproduction.

On the other hand, with a perceptual approach a plausible sound field is generated.

The signal processing is controlled by perceptual parameters such as source warmth, bril-

liance, early reflections to direct sound ratio, diffusion and late reverberation time. Even

though acoustic features implemented with this approach are usually derived from physical

properties of rooms, there is no precise simulation of an acoustic environment impulse re-

sponse. Its main applications are the creation of spatial effects for composers and plausible

reverberation units in music production. This approach is used in the Spat software [48]

and for sound environment modelling in MPEG-4 scene description language [117]. More

recently, in multimedia applications these two approaches have been used simultaneously

and even begin to overlap [116].

There are two main physical room acoustics modelling approaches, namely, geometrical

and wave-based methods. The most commonly used geometrical room acoustics modelling

methods are image-source and ray-tracing, which consist in decomposing the total sound

field into elementary waves and applying geometrical acoustics. Wave-based room acous-

tics modelling are based on solving the wave equation. These approaches include the

finite element method (FEM), the boundary element method (BEM), the finite difference

time domain method (FDTD), digital waveguide mesh (DWM), transmission line matrix

(TLM), and functional transformation method (FTM).


3.1.1 Geometrical Methods

In geometrical room acoustics, waves are replaced by sound rays. Such a simplification

may only be permitted when the dimensions of the room and wall surfaces are considerably

large compared with the wavelength, which is met for high frequencies only [64]. A unified

representation of geometrical room acoustics rendering algorithms is presented in [102],

and an integral room acoustics rendering equation is provided.

Ray-tracing is a geometrical method, where a finite number of rays are emitted from a

spherical sound source in all directions and all of them are traced in a sound environment,

as illustrated in Figure 3.1(a). The listener is represented by a spherical volume, and

rays which reach this volume are added to the output signal as reflections characterised

by energy, delay and diffraction [63]. In order to achieve statistically valid results, the

number of rays penetrating the volumetric listener object should be significantly large.

The basic specular reflection rule is commonly applied, where the incident angle of the

ingoing and outgoing rays are equal. More advanced techniques allow to incorporate

diffusion effects due to frequency-dependent scattering properties so that sound waves are

reflected in different directions from the boundary [69]. However, the drawback is that

such frequency-dependency might require separate ray tracing processes for each octave

band [110]. In addition, ray-tracing cannot handle edge diffraction effect caused by the

finite size of a boundary [110]. Furthermore, there is no guarantee that all the possible

ray paths from a source to a receiver will be found. In room acoustics modelling, this

technique is often applied to simulate higher-order reflections.

The image-source method is a ray-based approach, where all the reflected paths from a

source are replaced with direct ray paths from the mirror images of the source, as illustrated

in Figure 3.1(b). As for the output signal, the rays from image sources are delayed and

attenuated depending on the distance from the original source to a receiver [6]. Visibility

of each of the image sources has been studied in [13]. The image-source method finds

all the possible ray paths between a source and a listener. However, high computational

cost, which grows exponentially with reflection order, makes this method suitable only for

the first set of early reflections modelling. In reality, only a part of the reflected wave is

specularly reflected and a part is reflected in a diffuse fashion. The main drawback of this

method is that it cannot handle diffuse reflections. However, it can handle edge diffraction

by introducing edge sources for rigid walls [111] but no formulation is available to model

edge diffraction for impedance surfaces [110]. The image-source method is often used for

modelling early reflections in real-time rendering of virtual acoustic environments, in which

sound objects are moving [88]. Furthermore, a hybrid of the geometrical methods discussed

above has been developed in ODEON [78], in which early reflections are computed with

image-source technique and later reflections are handled with ray-tracing method. Similar


Source Receiver

Image source

Source

Receiver

(a) Ray-tracing method (b) Image-source method

Figure 3.1: he illustration of the general concept of geometrical methods: (a) image-source method,and (b) ray-tracing method.

hybrids have also been deployed in [85, 70].

There exist two other techniques in geometrical acoustics that are based on ray theory,

namely the beam tracing and radiosity methods. The beam tracing method is a derivative

of the ray-tracing algorithm in which rays are replaced by beams that are shaped like

unbounded pyramids. In this technique, beam-splitting occurs at boundary edges but it

cannot easily handle diffusion [31]. On the other hand, the radiosity method consists in

computing normal intensity at each wall element as contributions from source to the wall

element and between wall elements via so called form-factors [110]. The radiosity method

is only suitable for modelling diffusion as most versions of this technique do not model

specular reflections [110]. A combination of these two techniques can be found in [71].

3.1.2 Wave-based Methods

Wave-based approach to room acoustics modelling consists in numerical solving of the wave

equation. The analytical solution can only be found for idealised room shapes, such as

rectangular rooms with rigid walls. For the general case, a number of numerical techniques

exist for solving the wave equation by dividing the air volume or boundary surface into

elements. In the finite element method (FEM), the volume of the room is divided into

non-uniform elements at which the sound pressure is calculated [73]. In the boundary

element method (BEM) [56], only the boundaries of the space are discretised. The size

of elements determines the highest frequency for which the model is applicable. The

main advantages of element methods are the possibility to create a denser mesh for more

acoustically challenging structures and the ease of creating coupled models for propagation


in different media. However, the disadvantage of a finite element method is that extensive

pre-calculations are necessary to set up matrices describing arbitrary shapes of volume

elements. Furthermore, an inversion of this large matrix is required. Similarly to the

finite element method, the boundary element method that consists in approximating the

Kirchhoff-Helmholtz integral equation is very time consuming since all element-to-element

impulse responses have to be calculated. The complexity and hence computational cost

of both these element methods are considerably higher than in a finite difference method

[112, 110].

The FDTD can be treated as a special case of the finite element method in its time

domain formulation [110]. The finite difference method leads to a simpler algorithm for

solving the wave equation by approximating the time and space derivatives with finite

differences [109]. This technique is characterised by a regular spatial grid, the size of which

depends upon the sampling frequency. As will be explained in the following section, a

special case of the finite difference technique is a digital waveguide mesh, which has recently

been applied extensively to simulate sound propagation in rooms [87]. The concept of the

transmission line matrix (TLM) method is similar to digital waveguides in that it employs

networks of discrete transmission lines at scattering junctions [27, 11]. In the functional

transformation method, an initial boundary condition is first solved analytically searching

for eigenmodes of the modelled space, before it is discretised for computer simulations

[114]. Since the eigenvalues of an acoustic system can only be solved analytically for

simple shapes, splitting the modelled domain into separate blocks using the block-based

technique can be used to simulate the acoustics of complex-shaped rooms [81].

3.1.3 Motivation for the Chosen Method

A physical approach to room acoustics modelling is chosen for this research as it relates to

the natural human environment and is based on the use of physical parameters. Real-world

experiences have shaped the way in which people perceive and describe acoustic effects.

For example, an inexperienced user of reverberation effects based on perceptual approach

finds it difficult to describe the desirable sound effect in terms of perceptual parameters.

As for the choice of a physics-based method for further investigation, the wave-based

approach is more suitable as it accurately models physical acoustic effects. Among wave-

based methods, the FDTD method is selected for auralisation and architectural design

applications. This choice can be justified as follows.

Firstly, the ray-tracing method utilises a finite number of rays and assumes that the

reverberation is ideally diffuse. Consequently, some room acoustical defects such as flutter

echos might remain undetected in extremely long or flat enclosures [16, 86]. The image-

source method is not capable of modelling diffuse reflections; only specular reflections are


modelled where both the incident angles of the ingoing and reflected waves are equal. Most

importantly, however, in all geometrical methods the wavefront is represented by rays;

therefore, these techniques cannot accurately simulate wave-related phenomena such as

interference and diffraction. Due to the fact that geometrical methods do not approximate

the wave equation, they only offer a simplification that deviates significantly from the

actual physical solution, especially at low frequencies. This may lead to audible errors

in room impulse response modelling, and in turn reduced authenticity of the real room

acoustics.

Secondly, in comparison to finite differences, element methods have an advantage of

grid adaptability to irregular boundaries, which can be achieved by the use of non-uniform

shapes. However, this approach increases the computational cost, which effectively reduces

its application to very low frequencies only. Furthermore, the irregular grid in finite

element methods brings about a problem with moving sound objects in a modelled space.

That is, interpolation on coarse grid parts causes undesired numerical side effects. In the

boundary element method, each change of the position of a source requires time consuming

pre-computations.

Like element methods, the finite difference technique inherently models wave related

acoustic phenomena such as diffusion and interference. Straightforward algorithms in

finite difference schemes are computationally more efficient owing to the lack of heavy

pre-calculations in contrast to element methods. Furthermore, finite difference schemes

have the advantage of a regular spatial grid, which is beneficial for modelling of moving

sources and receivers, and preserving a constant valid frequency range. Consequently, in

this thesis, the finite difference time domain method is adopted for further research into

room acoustic modelling for audio and acoustic design applications.

3.2 The Finite Difference Time Domain Method

The finite difference time domain (FDTD) is numerical technique for solving the wave

equation. Thus this modelling technique is suitable for numerical simulation of wave prop-

agation in acoustic environments. In general, mesh based algorithms are applicable to the

simulation of whole sound fields, and consequently higher-order reflections modelling does

not increase the memory load. Moreover, room acoustic modelling with the use of a mesh

structure is correct in terms of timing and directionality of both the direct and reflected

signals, apart from some numerical dispersion errors at high frequencies. Furthermore, as

a method based on a mesh structure, wave-related phenomena such as interference and

diffraction are inherently modelled. This makes the method especially useful for sound

wave propagation modelling for low frequencies, where geometrical methods fail.

One of the main strengths of the FDTD method is that it is a time domain technique.


Therefore, when a wideband source signal is used as an excitation, such as a Gaussian

pulse, a wideband response of the system response can be obtained with a single simulation.

This feature is particularly useful in measurements of room impulse responses as the

resonance frequencies of rooms of complex shapes cannot be predicted analytically. It is

also useful to have a full bandwidth impulse response of an acoustic space for auralisation

purposes. Unfortunately, due to the fact that finite difference schemes suffer from a

direction-dependent dispersion error at high frequencies, the accuracy of the method is

usually reduced to frequencies much lower than the Nyquist frequency.

Numerical solving the wave equation with the use of the FDTD method consists in

approximating time and space derivatives with finite difference (FD) operators [112]. This

technique is usually characterised by a regular spatial grid, the size of which depends

upon the sampling frequency. The nonstaggered rectilinear standard leapfrog finite dif-

ference formulation of the wave equation [109] is obtained by applying centered difference

operators to approximate the derivatives in Equation (2.5). The name leapfrog actually

stems from the original staggered approach but it is now commonly used for an equivalent

nonstaggered implementation [109, 11]. Second-order accurate approximations applied to

discretise the 2D wave equation are

∂2p

∂t2=

pn+1l,m − 2pn

l,m + pn−1l,m

T 2+ O(T 2), (3.1)

∂2p

∂x2=

pnl+1,m − 2pn

l,m + pnl−1,m

X2+ O(X2), (3.2)

∂2p

∂y2=

pnl,m+1 − 2pn

l,m + pnl,m−1

X2+ O(X2), (3.3)

where X denotes the grid spacing and T is the time step. Assuming equal distances

between grid points in all directions, the 1D, 2D, and 3D discretised wave equations take

the form of Equations (3.4), (3.5), and (3.6), respectively [11]

pn+1l = λ2(pn

l+1 + pnl−1) + 2(1 − λ2)pn

l − pn−1l , (3.4)

pn+1l,m = λ2(pn

l+1,m + pnl−1,m + pn

l,m+1 + pnl,m−1)

+ 2(1 − 2λ2)pnl,m − pn−1

l,m , (3.5)

pn+1l,m,i = λ2(pn

l+1,m,i + pnl−1,m,i + pn

l,m+1,i

+ pnl,m−1,i + pn

l,m,i+1 + pnl,m,i−1)

+ 2(1 − 3λ2)pnl,m,i − pn−1

l,m,i, (3.6)


where λ denotes the Courant number, pnl,m,i is the pressure update variable, n is a time

index, and l, m, and i denote spatial indexes in x-, y-, and z-direction, respectively. Here,

pnl,m,i denotes a grid function representing an approximation to the solution p(x, y, z, t) to

the wave equation at position (x = lX,y = mX,z = iX) at time instance t = nT .

As will be shown in the next section, the Courant stability condition amounts to λ ≤ 1,

λ ≤ 1/√

2 and λ ≤ 1/√

3 for the 1D, 2D, and 3D scheme, respectively [11]. The Courant

number defines the relationship between grid spacing and the sampling frequency as

λ =cT

X, (3.7)

where c denotes a continuous-time wave speed, which for a given room conditions, such as

humidity and temperature, is considered constant. Hence for a given sample rate, Equation

(3.7) defines the distance between mesh nodes in the FDTD simulation according to

X =cT

λ. (3.8)

The FDTD technique requires that the entire computational domain (i.e., a physical region

over which the simulation is performed) is gridded. The grid spatial discretisation must

be sufficiently fine to resolve the geometrical features of the model and also the smallest

wavelength. Consequently, a large computational domain at a high sample rate requires

high memory and processing capacity, and results in a long simulation time.

3.2.1 Stability Analysis

The numerical system is considered stable if there exist no growing solutions of the system

[112]. The analysis of FD schemes establishes a numerical bound on the parameters that

occur in a given difference equation. It is possible to place a stability condition on an

FDTD scheme with the application of Fourier analysis, which in this context is referred

to as von Neumann analysis. With the use of a Fourier analysis, one can obtain both

necessary and sufficient conditions for the stability of finite difference schemes.

A complete von Neumann analysis consists in the application of the Fourier transform,

and hence on a presentation of Fourier integrals. However, a fully equivalent procedure

is to assume single frequency, plane wave solutions and investigate for which values of

the Courant number these solution can grow in time. Such an analysis is used here to

investigate the stability of a standard leapfrog scheme approximating the 1D wave equation

given by Equation 3.4, for which the single-frequency plane-wave solution reads

pnl = p0 ejωnT e−jklX , (3.9)


where p0 denotes the wave amplitude and k is the discrete-domain wavenumber. It is

sufficient to consider real-valued wavenumbers, i.e. the wavenumbers that lie in the range

−π/X ≤ k ≤ π/X. The classic relationship between the continuous-time domain and

z-domain is z = ejωT , and hence

pnl = p0 zne−jklX . (3.10)

Since the magnitude of z determines the rate of growth of the solution with time, z is in

this context referred to as an amplification factor [112]. By substituting Equation (3.10)

into (3.4), the following equation in z results

z + 2B(k, λ) + z−1 = 0, (3.11)

where

B(k, λ) = −1 + 2λ2 sin2(kX/2). (3.12)

The necessary stability condition is defined as |z| ≤ 1, which implies that the roots of

Equation (3.11) are located inside or on the unit circle [79] for all permissible values of k.

This condition is equivalent to

B(k, λ) ≤ 1, (3.13)

which places the following bound on the Courant number

λ2 ≤ 1

sin2(kX/2). (3.14)

We are actually looking for the minimum value of 1/ sin2(kX/2) that satisfies Equation

(3.14). Since sin2(kX/2) ∈ [0, 1], we only have to consider Equation (3.14) at its two

extrema, which yields the final stability condition of the standard leapfrog scheme ap-

proximating the 1D wave equation

λ ≤ 1. (3.15)

3.2.2 Dispersion Error

In reality, sound wave propagation in air is constant for all frequencies and in all propaga-

tion directions. When an acoustic system is simulated in a discretised domain, the phase

velocity of a sound wave differs slightly from the phase velocity of the modelled medium -

air. An unwanted side effect of using FDTD schemes is the numerical dispersion error, i.e.

the numerical phase velocity differs from the theoretical phase velocity, and for 2D and

3D rooms this phenomenon is generally dependent on frequency and propagation direc-

tion [11]. This numerical error is caused by the application of finite difference operators


and is dependent on the type of scheme used to approximate the wave equation. For all

FDTD schemes, the numerical phase velocity approaches the correct physical velocity at

low frequencies, whereas high frequencies generally travel at a lower speed than the real

sound wave propagation velocity. In particular, it is correct at DC (ω = 0) by consistency

with the wave equation.

The formula for the dispersion error is derived by substituting for centered finite differ-

ence operators in the respective discrete wave equation with the equations of the following

type

(pn+1

l,m − 2pnl,m + pn−1

l,m

)=

(z − 2 + z−1

)pn

l,m

= −4 sin2(ωT/2) pnl,m (3.16)

for the time derivative approximation. Similarly, the spatial approximations of a 2D

discrete wave equation can be represented as

(pn

l+1,m − 2pnl,m + pn

l−1,m

)=

(ejkxX/2 − 2 + e−jkxX/2

)pn

l,m,

= −4 sin2(kxX/2) pnl,m, (3.17)

in x-direction and

(pn

l,m+1 − 2pnl,m + pn

l,m−1

)=

(ejyxX/2 − 2 + e−jkyX/2

)pn

l,m,

= −4 sin2(kyX/2) pnl,m (3.18)

in y-direction, respectively. kx and ky denote the effective numerical wave numbers in both

directions of an acoustic space in an x-y plane, and the overall numerical wavenumber k

associated with wave propagation in the direction ω is given by

k2 = k2x + k2

x. (3.19)

Therefore, the directional numerical wavenumbers can be written in terms of the overall

numerical wavenumber k and the propagation angle θ as kx = k cos θ and ky = k sin θ.

Substituting Equations (3.16), (3.17), and (3.18) into the discrete 2D wave equation ap-

proximated with the standard leapfrog scheme the following dispersion equation results

1

λ2sin2

(ωT

2

)=[sin2

(Xk cos θ

2

)+ sin2

(Xk sin θ

2

)]. (3.20)

Similarly to the 2D case, the dispersion equation for the 3D leapfrog rectilinear scheme


can be derived with the use of analogous equations, which yields

1

λ2sin2

(ωT

2

)=

[sin2

(Xk cos θ cos φ

2

)

+ sin2(Xk sin θ cos φ

2

)

+ sin2(Xk sin φ

2

)], (3.21)

where θ and φ are the azimuth and elevation angles, respectively [112].

Rewriting Equation (3.20), the formula for the frequency in terms of the Courant

number, sample rate, propagation angle, and numerical wavenumber for a 2D standard

leapfrog scheme results

ω =2

Tarcsin

λ

√[sin2

(Xk cos θ

2

)+ sin2

(Xk sin θ

2

)] . (3.22)

As a measure of the dispersion error, the relative phase velocity is typically used. The

relative phase velocity is defined as a ratio between the numerical wave speed c and the

real sound wave propagation velocity c, and is given by the following equation

v(k, θ) =c

c=

ω

k c, (3.23)

where ω is computed from Equation (3.22).

In order to minimise the dispersion error, the Courant number is usually chosen at

the stability bound, i.e. λ = 1, λ = 1/√

2, and λ = 1/√

3 for modelling sound wave

propagation with the standard leapfrog scheme in 1D, 2D, and 3D rooms, respectively.

For the 1D case, the numerical dispersion actually vanishes for λ = 1. For the 2D and

3D standard leapfrog scheme, there always exists a direction in which the dispersion error

vanishes but in all other directions the numerical error is apparent. For instance, the

relative phase velocity of a 2D standard leapfrog scheme is illustrated in Figure 3.2. The

dispersion error is evidently strong in axial directions, whereas there is no dispersion in

diagonal directions.

3.2.3 Staggered FDTD Method

The FDTD method can alternatively be formulated on a staggered Yee’s grid. The classic

Yee’s method for solving Maxwell’s equations has been introduced in 1966 in [126]. Instead

of approximating the wave equation, the conservation of momentum given by Equation

(2.2) and the conservation of mass given by Equation (2.3) are approximated with centered


kx X

k y X

−π 0 π−π

0

π color bar

relative phase velocity0.7

0.75

0.8

0.85

0.9

0.95

1

Figure 3.2: Relative phase velocity for the 2D standard leapfrog scheme as a function of wavenumbersin x- and y-directions, respectively. The darkness in the plots indicates the relative phase velocity error,whereas white colour indicates a zero error.

finite difference operators. Consequently, a set of ordinary differential equations (ODE)

has to be solved for instead of one partial differential equation (PDE). In order to ensure

a second-order accuracy, all the applied finite difference operators have to be second-order

accurate.

By analogy to Section 3.2, we consider a 2D case for simplicity; the extension to 3D

is straightforward and is omitted here for brevity. The example of such an approximation

for a velocity component in x-direction reads [126, 112]

∂unx(l + 1

2 ,m)

∂t=

un+ 1

2x (l + 1

2 ,m) − un− 1

2x (l + 1

2 ,m)

T+ O(T 2). (3.24)

The unknown quantities in a basic staggered formulation of the FDTD approximation

using a Cartesian grid are pressure and particle velocity components. The acoustical

pressure pn(l,m) is determined at a position (x = lX,y = mX) at time instance t = nT .

The components of the particle velocity are determined at intermediate time instances

t = (n + 12 )T , and positions

(x = (l + 1

2)X,y = mX)

and(x = (l − 1

2)X,y = mX)

for

the velocity component in x-direction, and(x = lX,y = (m + 1

2)X)

and(x = lX,y =

(m − 12)X

)for the velocity component in y-direction, respectively. The Yee’s classic grid

is depicted in Figure 3.3.

Applying second-order accurate finite difference approximations to the conservation of


l X (l+1) X(l-1) X

m X

(m-1) X

(m+1) X

p u

u

x

y

ux

uy

Figure 3.3: Staggered Yee’s grid in 2D. Black-coloured circles indicate pressure node, grey-colouredand white-coloured circles indicate velocity components in x- and y-direction, respectively.

mass and conservation of medium equations, the 2D Yee’s scheme results

un+ 1

2x (l +

1

2,m) = u

n− 1

2x (l +

1

2,m)

− T

ρ X

[pn(l + 1,m) − pn(l,m)

]

un+ 1

2y (l,m +

1

2) = u

n− 1

2y (l,m +

1

2)

− T

ρ X

[pn(l,m + 1) − pn(l,m)

](3.25)

pn+1(l,m) = pn(l,m)

− ρ c2 T

X

[u

n+ 1

2x (l +

1

2,m) − u

n+ 1

2x (l − 1

2,m)

]

− ρ c2 T

X

[u

n+ 1

2y (l,m +

1

2) − u

n+ 1

2y (l,m − 1

2)],

(3.26)

where p denotes the acoustic pressure, ux and uy are particle velocity components in x-

and y-direction, ρ is the air density, and c is the sound velocity.

Concerning the dispersion of Yee’s staggered grid [126], the initial part of the derivation

of dispersion relation is similar to the analysis presented in Section 3.2.2. The acoustic

pressure and particle velocity components can be respectively represented as

Ux0

sin(

ωT2

)

T= P0

sin2(

Xk cos θ2

)

ρ X(3.27)


Uy0

sin(

ωT2

)

T= P0

sin2(

Xk sin θ2

)

ρ X(3.28)

P0 sin(ωT

2

)=

T

ρc2 X

[Ux0 sin2

(Xk cos θ

2

)+ Uy0 sin2

(Xk sin θ

2

)]. (3.29)

Substituting Equations (3.27) and (3.28) into (3.29), the dispersion relation of the Yee’s

staggered scheme results

1

λ2sin2

(ωT

2

)=[sin2

(Xk cos θ

2

)+ sin2

(Xk sin θ

2

)]. (3.30)

This equation is the same as the dispersion relation for the 2D standard leapfrog scheme

given by Equation (3.20). Consequently, the relative phase velocity can be computed

from Equation (3.23), where angular frequency is computed from Equation (3.22). Hence,

the dispersion of the 2D Yee’s scheme is illustrated in Figure 3.2, where the dispersion

is shown to be the strongest in axial directions and vanish totally in diagonal directions.

The stability condition of the 2D Yee’s staggered scheme is easily obtained by rewriting

Equation(3.30) in terms of the angular frequency and setting of the terms sin2(

Xk cos θ2

)=

1 and sin2(

Xk sin θ2

)= 1, which are their top possible values. Hence, the stability condition

is given as

cT ≤√

1

X2+

1

X2, (3.31)

which effectively gives the expected bound on the Courant number

λ ≤ 1√2. (3.32)

In conclusion, the staggered Yee’s scheme has been shown to have the same stability

condition and suffer from exactly the same dispersion as the standard leapfrog scheme.

This is well expected since both schemes solve for equations that are equivalent in the

continuous domain and both schemes are second-order accurate. Consequently, the nu-

merical solution of a simulation of room acoustics using these two FDTD methods based

on unstaggered and staggered grids is equivalent.

Botteldooren’s Boundary Model

The boundary model of a locally reacting surface for the staggered Yee’s scheme has been

proposed by Botteldooren in [15]. This simple frequency-dependent boundary model uses

mechanical impedance Zw(s) = R + Ms + Ks , where s = jω is the Laplace frequency

variable, R denotes resistance, K is the spring constant, and M denotes the mass per

unit area. The boundary condition in terms of acoustic pressure and particle velocity


components can be written as

p(t) = Run(t) + Mdun(t)

dt+ K

∫ t

−∞un(τ)dt, (3.33)

where un is the component of the velocity that is normal to the boundary plane. The

boundary formula based on the boundary condition given by Equation (3.33) is presented

here for the right boundary of a 2D system, parallel to the y-axis and located at a velocity

plane x = l+ 12 . Since the value of pn(l +1,m) lies outside of the modelled acoustic space,

an asymmetric finite difference approximation is used [15]

∂pn(l,m)

∂x= 2

pn(l + 12 ,m) − pn(l,m)

x(3.34)

that is only first-order accurate. The main advantage of the applied operator is that it

requires only one pressure value lying inside the modelled system and hence is computed

from the last equation of Equation (3.25). Since the value of acoustic pressure is not

defined at time instance t = nT , a linear interpolation for ux is used between the two

surrounding time instances, namely

unx(l +

1

2,m) =

1

2

(u

n+ 1

2x (l +

1

2,m) + u

n− 1

2x (l +

1

2,m)

). (3.35)

Finally, the update formula for the point lying on the boundary in terms of its previous

values and nearest pressure value is given as [15]

un+ 1

2x (l +

1

2,m) = α u

n− 1

2x (l +

1

2,m)

+ β2T

ρX−[pn(l,m) − KT

n∑

i=−∞u

i− 1

2x (l +

1

2,m)

], (3.36)

where

α =1 − R

ZF DTDT + 2MZF DTDT

1 + RZF DTDT + 2M

ZF DTDT

β =1

1 + RZF DTDT + 2M

ZF DTDT

(3.37)

ZFDTD =ρX

T,

where integration in the continuous domain is replaced with the summation up to now. As

for the stability of the Botteldooren’s boundary formulation, a more strict stability bound

is imposed on physical parameters of the boundary impedance than Equation (3.32). Such


Z-1

Z-1

JZ

-1

Z-1

Z-1

Z-1

J

Z-1

Z-1

Z-1

Z-1

Z-1

Z-1

JZ

-1

Z-1

Z-1

Z-1

Z-1

Z-1

JZ

-1

Z-1

Z-1

Z-1

Z-1

Z-1

Figure 3.4: 2D digital waveguide mesh structure. Each junction in the mesh is connected to fourneighbouring junctions with unit delays.

a stability condition for the boundary is given as [15]

CT

X≤

√√√√ 1 + 2MρX

1 + KXρc2

(3.38)

and it is more severe than Equation (3.32) if spring variable K is large compared to ρc/X.

The main disadvantage of Botteldooren’s boundary formulation is that its functionality

is constrained by a stability limit that depends on the physical parameters associated with

the boundary. Consequently, not all physically feasible parameter values can be used (e.g.

for a very high value of the spring constant and a very small value of the mass constant),

and thus not all real walls can be modelled.

3.2.4 Digital Waveguide Mesh

The starting point for the research into applying unstaggered finite difference schemes to

sound propagation modelling is the development of three-dimensional digital waveguide

mesh algorithm in 1994 [90]. The proposed structure is an extension of the two-dimensional

digital waveguide mesh (DWM) proposed by Smith and Van Duyne in [32].

The original digital waveguide mesh has a regular rectangular structure with each node

connected to its neighbours by unit delays, as illustrated in Figure 3.4. The scattering


junctions positioned at the regular node locations are interconnected with bi-directional

lines. At each junction, a conservation of energy and continuity law is imposed analo-

gous to Kirchhoff’s laws of power conservation in electrical circuit which apply to current

and voltage. In the context of room acoustics, the conservation of energy and power is

expressed in terms of sound pressure and particle velocity. If the digital waveguide net-

work is constructed from waveguide elements in a regularly arranged grid, the network of

scattering junctions is called the digital waveguide mesh. This technique has been widely

applied in artificial reverberation [104, 50] and sound synthesis [121, 119]. Each delay

element is bidirectional and the sound pressure of the waveguide is given by the sum of

pressure values associated with waves travelling in both directions and indicated here as

input (p+i ) and output (p−i ) waves after [89]

pi = p+i + p−i . (3.39)

There are two conditions (that follow from the main two acoustic laws discussed in Section

2.1) that apply to a lossless scattering junction with N neighbouring bidirectional delay

lines. The first is that the flow adds to zero, i.e. flow is the sum of ingoing velocities

equals the sum of outgoing velocities. The second condition is that the sound pressure is

equal at the output port of the junction. With the use of these two conditions, the sound

pressure at a scattering junction can be expressed as

p =2∑N

i=1p+

i

Zi∑Ni=1

1Zi

, (3.40)

where Zi denotes waveguides impedance defined as Zi = pi/ui. The input of one end of

the bidirectional connection equals the output of the second end of the waveguide. This

can be expressed as

p+i = z−1 p−i,opp, (3.41)

where p−i,opp represents the second end of the waveguide i. Note that impedances in all

directions are equal in a homogeneous medium. Combining Equations (3.40) and (3.41)

yields the update formula [89]

p =z−1 2

N

∑Ni=1 pi,opp

1 + z−2, (3.42)

where N defines the number of neighbouring junctions. Since the physical variables are

separated into directional wave components, such an implementation is in a DWM litera-

ture referred to as wave-variable digital waveguide mesh (W-DWM) [49].

An alternative implementation using Kirchhoff variables is referred to as Kirchhoff-


Z-1

Z-2

p zk-1

p zw-1

p w+

p w-

Figure 3.5: Filter block of a KW-pipe for interfacing the Kirchhoff (left) and wave (right) variableimplementations.

variable digital waveguide mesh (K-DWM), by analogy to Kirchhoff’s integral solution of

Maxwell’s equations. In fact, its implementation is the same as FDTD implementation

where the Courant number is set at its top value. Therefore in the mathematical sense,

the digital waveguide mesh is considered an explicit finite difference scheme. For instance,

the rectilinear digital waveguide mesh implemented with Kirchhoff variables is a standard

leapfrog scheme.

The combination of the K-DWM and W-DWM approaches is possible with the use

of so called KW-pipes which have been introduced in [49]. These converters, illustrated

in Figure 3.5, allow for the conversion of wave variables into Kirchhoff variables and vice

versa, and can be applied to combine two types of implementation in one simulation.

Even though KW-pipes can be applied to multiport scattering junctions, the formulations

presented in the DWM literature (such as in [49, 53, 75]) allow for 1D conversion only.

In the context of room acoustics, these have been often applied to combine K-DWM

implementation of the room interior with 1D reflectance filters at a boundary, e.g. in

[53, 75].

One obvious advantage of K-DWM (and also FDTD) implementation is that it requires

less arithmetic operations and two times less memory than W-DWM. This is due to the

fact that one delay is used per node instead of two delays in bidirectional waveguides of

the wave-variable implementation. Consequently, the examples of the digital waveguide

mesh based on a rectilinear mesh are presented in this section in their K-DWM form. In

fact, this implementation is commonly used in room acoustics simulations using digital

waveguide mesh because of computational efficiency, e.g. in [93, 53, 75].


The Equivalence of DWM and FDTD Methods

The equivalence of the digital waveguide mesh and the finite difference time domain meth-

ods can be shown with the example of the discrete-time solution of the 1D wave equation,

in which we largely follow the derivation presented in [106]. The 1D wave equation in a

scalar form reads∂2u

∂t2= c2 ∂2u

∂x2(3.43)

and we are looking for its possible solution u(x, t). If we consider the d’Alembert solution

of the wave equation which is based on the propagation of two waves in opposite directions

according to

u(x, t) = u+(t − x

c) + u−(t +

x

c), (3.44)

where u+ and u− are right-going and left-going travelling wave components. In the context

of acoustics, these variables represent the displacement of waves travelling in the positive

and negative x-directions, respectively. The discrete version of Equation (3.44) reads

u(lX, nT ) = u−(nT − lX

c) + u−(nT +

lX

c)

= u+[(n − l)T ] + u−[(n + l)T ], (3.45)

where substitution X = cT in Equation (3.45) results from the stability condition λ =cTX = 1. In order to produce a physical output of a 1D system, the left- and right-going

travelling wave components are summed to

u(l, n) = u+(n − l) + u−(n + l). (3.46)

The 1D leapfrog scheme equation is given by Equation (3.4), which written in an analogous

form to Equation (3.46), reads

u(n + 1, l) = u(n, l + 1) + u(n, l − 1) − u(n − 1, l). (3.47)


If we now expand the right hand side of Equation (3.47) using analogous formulae to

Equation (3.46), one obtains

u(n, l) = u(n, l + 1) + u(n, l − 1) − u(n − l, l)

= u+(n − l − 1) + u−(n + l + 1)

+ u+(n − l + 1) + u−(n + l − 1)

− u+(n − l − 1) − u−(n + l − 1)

= u+(n − l + 1) + u−(n + l + 1)

= u+[(n + 1) − l] + u−[(n + 1) + l], (3.48)

which shows that the solutions of the digital waveguide mesh and the finite difference time

domain method are equal. The equivalence has also been shown in 2D and 3D [49].

Rectilinear Digital Waveguide Mesh

In a 2D case, each node of the rectilinear Kirchhoff variable digital waveguide mesh is

updated according to the formula [89]

pn+1l,m =

1

2(pn

l−1,m + pnl+1,m + pn

l,m−1 + pnl,m+1) − pn−1

l,m , (3.49)

where pn+1l,m is the updated node, l and m denote spatial indexes, and n denotes a time

index for neighbouring points. Therefore, two previous values are necessary to compute the

pressure value at the next time, which can be effectively implemented using two pressure

variables per grid point. Note that Equation (3.49) is the same as Equation (3.5) for

λ = 1√2. Thus such an implementation is equivalent to the implementation of the 2D

standard leapfrog scheme.

As far as the dispersion characteristic is concerned, the original 2D and 3D digital

waveguide mesh suffers from a direction-dependent frequency error [105]. Figure 3.6(a)

shows the relative phase velocity as a function of numerical wavenumbers for the 2D

rectilinear DWM, in which the wave propagation speed is shown to be constant only in

diagonal directions. In axial directions, on the other hand, the propagation speed decreases

when frequency increases. This plot is identical to Figure 3.2, which confirms that the

rectilinear digital waveguide mesh is equivalent to the standard leapfrog scheme at the top

value of the Courant number.

With the original digital waveguide mesh structure, phase errors are quite severe and

this method produces accurate results for 2D and 3D room acoustics simulations only

up to one tenth of the sampling frequency [90]. Alternative sampling lattices have been

found to have better dispersion error characteristics as the error is spread more uniformly


kx X

ky X

−π 0 π−π

0

π

kx X

ky X

−π 0 π−π

0

π color bar

relative phase velocity0.7

0.75

0.8

0.85

0.9

0.95

1

(a) (b)

Figure 3.6: Relative phase velocity for (a) the rectilinear digital waveguide mesh and (b) the interpolateddigital waveguide mesh, as a function of x- and y-direction wavenumbers. The darkness in the plotsindicates the relative phase velocity error, where white indicates a zero error, and any error larger thenor equal to 0.3 is represented with black colour.

in all directions. The triangular mesh lattice is proposed as a main alternative to the

rectangular mesh due to nearly direction-independent dispersion error in [38]. However,

mesh structures that are not based on a rectilinear grid are inconvenient for modelling

rectangular shapes commonly occurring in real rooms. Therefore, in order to make the

error of the rectangular mesh homogeneous in all directions, interpolation techniques have

been introduced.

Interpolated Digital Waveguide Mesh

In the original digital waveguide mesh structure, each junction is connected only with

axially neighbouring nodes. In order to reduce the dispersion error, each junction is

connected with all the diagonal and axial neighbours. Interpolation makes the dispersion

error nearly independent of the wave propagation direction [93]. The difference equation

for the 2D interpolated digital waveguide mesh can be written in the finite difference form

as [11]

pn+1l,m = λ2[a(pn

l−1,m + pnl+1,m + pn

l,m−1 + pnl,m+1) +

b(pnl−1,m−1 + pn

l+1,m−1 + pnl−1,m−1 + pn

l+1,m+1) + cpnl,m] − pn−1

l,m , (3.50)


where the optimised coefficients are given as

a = 0.6241,

b = 0.5(a − 1),

c = 2λ2 − 4(a + b),

λ = 1√2.

(3.51)

Both first- and second-order linear interpolations have been applied to find the weighting

coefficients. However, the best result in terms of directionally independent error has

been obtained for the optimally interpolated digital waveguide mesh [91]. The optimised

parameters are obtained from minimising the relative frequency error [92]. The relative

phase velocity is illustrated in Figure 3.6(b) and it can be seen that for the valid frequency

range, which is limited to a half of the Nyquist frequency, the dispersion error is equal in

all propagation directions. Furthermore, such a directionally independent dispersion error

is the same as for the rectilinear digital waveguide mesh in axial directions.

These results are very similar to an interpolated approach applied in finite difference

time domain schemes, where the smallest dispersion is obtained by superimposing two

rectilinear schemes, one of which is rotated by 45 degrees [115]. The finite difference

scheme can be then optimised by adjusting one free parameter to distribute a propagation

speed equally in all directions [11]. Despite the fact that dispersion in the interpolated

digital waveguide mesh is directionally independent, the error still remains. For accurate

room acoustics modelling, such a substantial dispersion can be a source of significant

errors. Therefore, the use of frequency warping technique has been proposed in order to

reduce the direction-independent dispersion error [93]. Note that the frequency warping

technique can only be applied if the dispersion is directionally-independent.

Boundary Models

Boundary modelling is fundamental for accurate room acoustics modelling. In particular,

there is a necessity for controllable boundary conditions, where the reflection coefficient

of the wall or other boundary could be precisely represented in a simulation. The vast

majority of the boundary formulations for the digital waveguide mesh simulations of room

acoustics are based on a 1D mesh termination, illustrated in Figure 3.7. Typically, the

discrete boundary is modelled by introducing extra boundary nodes and interconnecting

them with the edge nodes of the simulated mesh, which represent the interior of a room.

The idea is that only the boundary node and the adjacent node in the room interior space

are needed in order to derive the local velocity component normal to the wall. This way,

wave propagation is locally assumed to be one-dimensional at a boundary, hence the model

does not implement wave propagation along boundaries. This simplification is somewhat


Z-1

Z-1

Z-1

Z-1

Z-1

Z-1

J

Z-1

Z-1

Z-1

Z-1

Z-1

Z-1

JZ

-1

Z-1

Z-1

Z-1

Z-1

Z-1

B

B

D

D

Figure 3.7: 1D boundary termination of the digital waveguide mesh using a dummy node.

unphysical, but for high impedances it still roughly approximates locally reacting surface

theory.

1D Frequency-independent Boundaries

The first boundary of this kind was introduced by Savioja et al. in [90] for the original

digital waveguide mesh that is implemented as an FDTD model, thus using Kirchhoff

variables. The basic approach is to calculate the node values with 1D formula

pB(n) = (1 + R)p1(n − 1) − RpB(n − 2), (3.52)

where R represents a reflection coefficient in the range of −1 < R < 1, p1 is the discrete

sound pressure in front of the boundary one time step ago and pB is the discrete sound

pressure at the boundary two time steps ago [90]. Applying this basic solution to multi-

dimensional digital waveguide mesh proved to be correct for high values of the reflection

coefficient and low angles of incidence. However, due to the change in dimensionality,

undesirable reflections occur for small values of the reflection coefficient.

An improvement to these unwelcome reflections for relatively low real impedance values

has been made by applying Taylor series approximations in [77]. Better performance was

noticed for low values of the reflection coefficient when the first-order solution of the series

is applied. Such a boundary formula designed to improve the case for R = 0 is given by


[77]

pB(n) = 2p1(n − 1) − (1 − R)p2(n − 2) − RpB(n − 2). (3.53)

This approach is still one-dimensional and as such does not provide a general solution

for multidimensional boundary modelling. The improvement is only noticed for positive

values of r and a significant error occurs at the low frequencies, even for high values of

the reflection coefficient. In addition, the performance of such a boundary for high angles

of incidence is not improved. Further enhancement has been proposed with the use of a

second-order FIR filer but this makes the simulation valid for an even narrower frequency

band [54].

Frequency-independent Nonlocally Reacting Surface Model

More recently, the first attempt to model nonlocally reacting walls in DWM simulations

has been proposed in [51]. The change of admittance of this multidimensional approach

is incorporated at the end of the simulated interior of the room for reflective boundary

modelling. The admittance Y can in general be designed separately for each connection

of junctions according to

Y =1 − R

1 + R, (3.54)

and the admittance layer should consist of at least 4 rows of mesh points in order to

obtain decent results and ensure stability. Part of the signal gets through the admittance

boundary and reaches the absorbing boundary located behind the former one, which trun-

cates the mesh. The propagated signal is absorbed (at the absorbing boundary) using the

spatial filter-based absorbing boundary condition [55]

pB(n) = hBa1(n − 1) +hBd1

2[p1,x−1(n − 1) + p1,x+1(n − 1)] +

hBd2

2[p2,x−1(n − 2) + p2,x+1(n − 2)]

hBa2p2,x(n − 2) + hBa3p3,x(n − 3), (3.55)

where subscript B designates the node coordinates taking into account three neighbours

perpendicular to a boundary and four diagonal neighbours (two per each direction), and

h represents axial and diagonal filter coefficients, respectively. The coefficients of the filter

have been optimised numerically using a Nelder-Mead algorithm for the 2D rectangular

digital waveguide mesh. Therefore, this termination is not directly applicable in other

mesh topologies or 3D meshes. This nonlocally reacting boundary structure performs well

for positive values of the reflection coefficient but still some error is encountered due to

a nonoptimal absorbing boundary [51]. Nevertheless, this technique usually gives better

results than the 1D boundary termination.


The idea of modelling boundaries as nonlocally reactive surface has also been applied

to the 3D DWM in [52]. The motivation for this boundary model was unacceptably high

numerical error in reflectance that resulted for 3D simulations with the use of aforemen-

tioned 1D boundary terminations [52]. However, the admittance boundary in a 3D case

did not produce good results due to the lack of a highly absorbing termination of a 3D

homogeneous boundary layer. Hence, a 1D termination was used for that purpose. In

order to obtain a 3D boundary model for any reflection coefficient, a 3D highly absorptive

boundary model would be required.

1D Frequency-dependent Model with KW-pipes

The aforementioned methods for modelling boundaries in K-DWM only allow modelling

a real-valued reflection coefficient that is constant over all frequencies. However, the re-

flectance of real boundaries is generally frequency-dependent. As material parameters are

given for each octave band, the simulation would have to be run for each octave separately.

This approach can be extended to frequency-dependent boundaries by replacing the re-

flection coefficient with a digital filter [46]. Whereas the termination of a room interior

modelled with wave variable digital waveguide mesh is completely straightforward, such a

simulation would require much more resources than the Kirchhoff variable room interior

implementation. Therefore, the use of KW-pipes in room acoustics has been proposed

to combine the K-DWM room interior implementation with the reflectance filters at a

boundary in [53] and [75]. Note that the boundary termination with the use of KW-pipes

is always one-dimensional due to the nature of KW-converters that are applied between

the K-DWM node and W-DWM junction. This also applies to multidimensional simu-

lations in which a dummy junction (that passes waves travelling in both directions) is

introduced and to which the reflectance filter is connected such as in [53] and [75]. It can

be concluded that the frequency-dependent boundary terminations available in the DWM

literature are always based on the 1D termination, and hence suffer from the same prob-

lems as frequency-independent boundary models. Therefore, at nonperpendicular angles

of incidence their performance is significantly deteriorated [37].

Modelling Diffusion

In addition to specular reflections, methods to model diffuse reflections in the DWM have

been studied. For example, basic quadratic residue diffusers were modelled with delay

lines of integer length in [68, 100]. Unfortunately these models also use 1D reflection

coefficients to model the boundary, an approach that may lead to significant boundary

reflection errors. Most other studies in the DWM field tend to aim towards controllable

diffusion rather than predictivity. For example, in [67] circulant matrices were applied,


which result in random redirection of reflected waves at the boundary. A similar approach

has been taken in [101], where in order to avoid the rotation error caused by the limited

number of directions, it was proposed to apply the rotations at the mesh nodes adjacent to

the boundary, resulting in a so-called “diffusing layer”. While high levels of diffusion can

be induced with this approach, a downside is the unphysicality of placing a local, point-

based scattering mechanism in front of the actual boundary. The implications are that

the performance depends strongly on the mesh topology, and the model is unpredictable

for high angles of incidence when sound waves can be rotated more than twice or not even

once [101].

Whereas the aforementioned diffusion models are associated with boundaries, an al-

ternative approach is to place diffusion elements in the interior of the waveguide mesh,

which can be viewed as modelling objects within the acoustical space [4].

3.2.5 Implicit Methods

The methods described so far in this chapter can be classified as explicit finite difference

methods, i.e. the discrete solution to the wave equation in a given mesh node is computed

explicitly from discrete values of this and neighbouring grid points at previous time steps.

Such explicit compact finite difference schemes, which look at the nearest neighbouring

points in discrete space domain, are only second-order accurate in both time and space

domain. Higher-order accuracy schemes are difficult to design mainly due to providing

stability near boundaries and low operation count. The improvement in accuracy of the

FDTD method is possible with the use of implicit schemes, which can even be fourth-

order accurate in time and space. Therefore, an implicit finite difference method seems

an interesting choice for room acoustics modelling, especially that it achieves the highest-

order of accuracy on the smallest mesh system [19]. Furthermore, implicit methods usually

have better stability properties than explicit schemes based on the same spatial stencil.

Consequently, the accuracy of such a method is higher for a wider frequency range, relative

to the sampling frequency. It is even possible to derive an unconditionally stable compact

implicit scheme approximating the wave equation, which cannot be achieved with the use

of compact explicit finite difference schemes.

The general update equation for a node in a compact implicit FD scheme approximat-

ing the 2D wave equation can be written as

[1 + a(δ2x + δ2

y) + d(δ2xδ2

y)]pn+1l,m = [2 − b(δ2

x + δ2y) − e(δ2

xδ2y)]p

nl,m −

[1 + c(δ2x + δ2

y) + f(δ2xδ2

y)]pn−1l,m , (3.56)

where δ2x and δ2

y denote second-order difference operators approximating the derivatives

in space, respectively. Unfortunately, multidimensional implicit methods require solving


a set of equations at an advanced time level [constructed from terms on the left hand

side of Equation (3.56)]. Therefore, the problem of inversion of a multidimensional matrix

arises. Such matrix inversion algorithms are in general computationally very demanding,

especially for a matrix without a simple pattern that would result for modelling acoustic

spaces of complex shapes.

To address this problem, alternating direction implicit (ADI) methods have been in-

troduced [30], in which the problem of inverting a multidimensional matrix is reduced to

a succession of many one-dimensional inversion problems by factorising the scheme [35].

Each of these subequations involves an inversion of a one-dimensional matrix only, which

consists in finding the solution of a three-diagonal set of equations. The rest of terms in

these equations can be computed explicitly from the values stored in the memory.

Some of the splitting formulae for the 2D compact implicit scheme implemented with

the use of an alternating direction implicit method include the following formulations [42]

{(1 + aδ2

x)pn+1∗l,m = λ2[(δ2

x + δ2y) + bδ2

xδ2y ]p

nl,m,

(1 + aδ2y)(pn+1


l,m ) = pn+1∗l,m ,

(3.57)

and {(1 + aδ2

x)pn+1∗l,m = λ2

a [−1 + (a − b)δ2y ]pn

l,m,

(1 + aδ2y)(p

n+1l,m − 2pn

l,m + pn−1l,m ) = pn+1∗

l,m + λ2

a (1 + bδ2y)pn

l,m,(3.58)

where pn+1∗l,m denotes an intermediate value that is computed from the first equation. Equa-

tion (3.58) has been identified in [42] as the most efficient computationally since it involves

the calculation of δ2y only at each node. In all splitting formulae, the global accuracy of the

method is preserved only if the intermediate boundary conditions are obtained explicitly

from the second equation. That is, the intermediate boundary condition in a splitting

given by Equation (3.58) is calculated from

pn+1∗l,m = (1 + aδ2

y)(gn+1 − 2gn + gn−1) − λ2

a(1 + bδ2

y)gn, (3.59)

where gn denotes an explicit value of the boundary node computed from the respec-

tive boundary condition which in the continuous domain can be expressed as p(x, y, t) =

g(x, y, t).

Adjusting the scheme’s free parameters may minimise the problem of the overall dis-

persion error or the directional dependence of numerical dispersion [12]. An interesting set

of parameters has been proposed in [35], for which a fourth-order accurate scheme results.

This is the only compact finite difference scheme that offers such a high accuracy on the

smallest mesh system.

There is a convenient algorithm to inverse the three-diagonal system of linear equations


A

B C

D

(a) (b)

Figure 3.8: The implementation procedure of the alternating direction implicit method in (a) a rectan-gular room and (b) an L-shape acoustic enclosure.

that arise in alternating direction implicit technique, namely, the Thomas algorithm [109].

This technique allows for a fast implementation of 1D matrix inversion, and consequently

the computational load is not much increased compared to explicit methods. A detailed

explanation of the Thomas algorithm is presented in Section 3.4.

ADI Implementation

Let us here briefly outline the implementation of computational procedure for ADI meth-

ods in a 2D domain. The implementation of a rectangular space, depicted in Figure 3.8(a),

is very straightforward and is based on solving for discrete pressure values lying along the

lines in x-direction according to the first equation. This way intermediate values are ob-

tained in all grid points of the simulated room interior. Such a procedure is repeated

for all horizontal lines in a larger loop which goes through the mesh lines in the vertical

direction. Secondly, the second equation is used to calculate the pressure values along the

vertical mesh lines using past pressure values and intermediate values computed at the

first step. This line by line calculation is again done in a larger loop that goes through

the mesh lines in the horizontal direction [109].

When the simulation region is nonrectangular, the procedure for the implementation

of the ADI technique is analogous to the disintegrating operator technique given in [34].

As an example, the L-shaped region, illustrated in Figure 3.8(b), is considered. At first,

the intermediate values pn+1∗l,m at all points of the modelled space are determined with the

use of the first equation of (3.58) apart from grid point B. The intermediate value at B

cannot be calculated since it depends on the value of pn+1l,m at the point D which is yet to

be computed [36]. Thus all the values along horizontal mesh lines can be computed apart


from the mesh line BC. Next, the value of pn+1l,m is computed at mesh point D by solving

the second equation of (3.58) along the mesh line AB. Having obtained pn+1l,m at D, we

calculate an intermediate value at B. Next, the missing line in the horizontal direction BC

is determined with the first equation. Finally, since all the intermediate values are known,

the computation along all vertical lines can be done using the second equation of (3.58).

Note that the mesh line AB has already been determined. This procedure may easily be

applied to ADI methods in general regions in 2D and 3D acoustic spaces [36]. However, it

requires a special treatment of inner corners, also referred to as reentrant corners, which

for complex geometries of the modelled space may severely complicate the implementation

of the ADI method.

3.3 Frequency Warping

The frequency warping technique [80] consists in pre-processing the input and post-

processing the output signals of the simulation with a warped FIR filter, which reduces

frequency shifting to compensate for the dispersion error [93]. The frequency warping

technique can be applied to reduce the dispersion error in methods, for which dispersion

is directionally independent. Reducing the direction-independent numerical error extends

the frequency range in which the simulation is valid. Such a correction is considered to

be correct as the difference between the maximum and minimum error after warping is

relatively small [91].

A frequency warping FIR filter structure with a unit delay replaced with a first-order

allpass filter is depicted in Figure 3.9. The transfer function of such an allpass filter is

given by

H(z) =z−1 + α

1 + αz−1. (3.60)

The extent of warping depends on the value of a coefficient α, which is constant for all

the filters. The tap coefficients are set to input signal samples, the one to be warped. An

impulse is fed into the FIR filter and the output of the FIR filter is a frequency-warped

version of the original signal. In addition to warping, changing the sample rate allows

for nonlinear frequency shifts, which are impossible to obtain by just using the warping

technique. A number of successive first-order frequency warping and resampling operations

is advised in order to reduce the error significantly. With the use of multiwarping, the

dispersion error of the 3D interpolated waveguide mesh is reduced to 1.2% [94].

Frequency warping can also be implemented in the frequency domain by nonuniform

resampling of the Fourier transformed signal [103]. Such a Fourier transform has to be

performed on a large number of data points in order to maximally reduce the artefacts of

the interpolation of spectral data. As the frequency response is in general complex-valued,


H(z)δ(n)

y(n)

x(1)x(0) x(2) x(N-2) x(N-1)

H(z) H(z) H(z)

Figure 3.9: A frequency warping FIR filter structure.

interpolation should be applied to both real and imaginary parts so that both phase and

magnitude are preserved [95].

The great advantage of this method is that the cost of warping does not depend on

mesh size. However, the computational cost of this method depends on the length of the

warped signal and it is only applicable to finite input signals. Hence the main disadvantage

of the frequency warping technique is that pre- and post-processing has to be executed

off-line, which makes this method unsuitable for interactive audio applications.

3.4 Solving Tridiagonal Systems

The alternating direction implicit method is based on splitting the multidimensional ma-

trix inversion problem into a 1D matrix inversion problem. Hence, an efficient method to

inverse a 1D tridiagonal matrix is required. A convenient algorithm to solve tridiagonal

systems of equations which arise in implicit finite difference schemes, called Thomas al-

gorithm, is presented in this section. The Thomas algorithm is a simplified form of the

Gaussian elimination that can generally be used to solve tridiagonal systems of equations

[109].

Consider the following system of equations for l = 1, 2, ..., L − 1

alpl−1 + blpl + clpl+1 = dl, (3.61)

where al, bl, and cl are coefficients, pl denotes the unknown pressure value to be computed

at the time step n + 1 at the position denoted with subscript l. The right hand side of

the equation that can be computed explicitly is denoted as dl. Such a tridiagonal system

of equations written in the matrix form yields

b1 c1 0

a2 b2 c2

. . . . . . . . .

aL−1 bL−1 cL−1

0 aL bL

p1

p2

. . .

pL−1

pL

=

d1

d2

. . .

dL−1

dL


In order to solve for three pressure values at the next time step with the use of the

Thomas algorithm, we need to know the pressure value at the left and right boundary

p0 = β0, pL = βL, (3.62)

which have to be computed explicitly before the tridiagonal system can be solved.

Firstly, the following expression is applied to replace Equation (3.61)

pl = el+1pl+1 + fl+1, (3.63)

where e and f are auxiliary 1D vectors for l = 0, 2, ..., L − 1, the values of which are to

be determined. Substituting Equation (3.63) for pl+1 into Equation (3.61), the following

expression results

al(el+1pl+1 + fl+1) + blpl + clpl+1 = dl. (3.64)

Equation (3.64) is consistent with Equation (3.61) if the following conditions are met

el+1 = −clgl (3.65)

fl+1 = (dl − alfl)gl, (3.66)

where gl is given as

gl = (al + blel)−1. (3.67)

Thus, for known initial values of e1 and f1, the computation of el and fl is straightforward

from Equations (3.65) and (3.66) for index l greater than 1. At the left boundary, the

boundary condition p0 = β0 is consistent with p0 = e1p1 + f1 for the following initial

auxiliary values e1 = 0 and f1 = β0. The second part of the Thomas algorithm consists

in solving system using Equation (3.63). Note that the solution for the node at a right

boundary is known from the right boundary condition pL = βL.

The Thomas algorithm to solve the tridiagonal system given by Equation (3.61) with

the values at the boundary given by Equation (3.62) can be summarised as follows. Firstly,

e1 = 0 and f1 = β0 are initialised. Secondly, the values of auxiliary vectors are computed,

in a loop on l from 1 to L − 1, according to Equations (3.65) and (3.66). Finally, within

the loop on l from L − 1 to 0, the solutions of the tridiagonal system of linear equations

are calculated according to Equation (3.63).

3.5 Fractional Delays

One of the goals of this thesis is to implement diffusers based on a phase grating approach

by adding delays to reflectance filters, as will be explained in Chapter 8. In general,


phase grating diffusers may require arbitrary well depths that do not fall on the linear

grid. Therefore, interpolation techniques are necessary to simulate a delay of a noninteger

multiple of the sample period, the so called fractional delay.

Such noninteger length delays are often applied for interpolation between samples in

delay lines smoothly varying their length in order to correct the difference between the

desired time delay and the nearest multiple of the sample interval. Arbitrary digital delays

are also commonly used in sound synthesis, e.g. in a lossless feedback loop in a vibrating

string simulation [106] and in classical lossless tube model that concatenates tube sections

of different diameters [118].

An ideal fractional delay element (i.e., an ideal band limited interpolated) is nonre-

alisable since the corresponding impulse response would be noncausal and infinitely long

[118]. Therefore, approximation techniques have to be applied, the most commonly ap-

plied kinds of fractional delay filters being: Lagrange FIR interpolation filters and Thiran

allpass interpolation filters.

Both of these techniques yield maximally flat group delay at low frequencies and com-

putational cost of both implementations approximating an ideal fractional delay is compa-

rable [118]. A linear Lagrange interpolation FIR filter has an advantage of a nonrecursive

implementation which does not introduce any significant numerical problems for time-

varying coefficients. However, it suffers from amplitude attenuation at high frequencies.

Such a lowpass filtering effect is often undesired, and consequently its applicability is

sometimes limited to heavily oversampled cases. Conversely, allpass filters are filters of a

unity amplitude for all frequencies and thus a Thiran allpass filter is a more commonly

used interpolator in the field of audio. The transfer function of a discrete-time allpass

filter is given as [118]

A(z) =aN + aN−1z

−1 + ... + a1z−(N−1) + z−N

1 + a1z−1 + ... + aN−1z−(N−1) + z−N, (3.68)

where N denotes the order of the filter and ai are real filter coefficients. Thiran has

originally proposed a method for calculating optimally flat group delay response at low

frequencies for an all-pole IIR filter, which suffered from the lack of control of the mag-

nitude response. This method has been extended to designing allpass filters, resulting in

the following coefficients of a maximally flat group delay allpass filter [66]

ak = (−1)k

(N

k

)N∏

n=0

△− N + n

△− N + k + n, k = 0, 1, 2, ..., N (3.69)


(a)

(b)

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

Normalised frequency

Ph

ase

de

lay

[sa

mp

les]

0 0.1 0.2 0.3 0.4 0.5

1

1.5

2

2.5

3


Ph

ase

de

lay

[sa

mp

les]

Figure 3.10: The phase delay of a Thiran allpass filter of a: (a) first-order and (b) second-order,respectively. The dashed lines indicate an ideal fractional delay.

where the k binomial coefficient is given as

(N

k

)=

N !

k!(N − k)!(3.70)

and △ denotes the value of the fractional delay. Note that the delay parameter △ refers

to the actual delay length rather than the offset from N samples. The Thiran design

technique is the only closed-form solution known for designing allpass interpolators of

arbitrary order [106]. Since the filter nominator is an inverse version of the denominator

polynomial, the poles and zeros of the filter have the same angles in the complex plane

but inverse radii, and hence the magnitude response of the filter is perfectly flat at 1 for


^N samples delay Z

-1y(n-N- )

-a1

a1

y(n)

First-order Thiran allpass filter

Figure 3.11: A diagram depicting a delay line in series with a first-order Thiran allpass filter.

any coefficients [118]

|A(ejω)| =

∣∣∣∣e−jωN △ (e−jω)

△(ejω)

∣∣∣∣ = 1, (3.71)

and a group delay, which is most accurate at low frequencies, is given as [118]

τA(ω) = −∂θA(ω)

∂ω= N − 2τ△(ω), (3.72)

where τ△(ω) is the group delay of 1/ △ (ejω). The phase delay (in samples) for a number

of desired delays at ω = 0 for the first-order and second-order Thiran allpass filters is

illustrated in Figure 3.10(a) and (b), respectively. The magnitude plots are not provided

as the amplitude response is 1 at all frequencies. As depicted in Figure 3.10, the Thiran

allpass filter matches perfectly a required phase delay at ω = 0. However, the phase delay

approximation is less correct at high frequencies, eventually converging to the delay value

given by the filter order.

Note that since a0 = 1, such an allpass filter does not require further scaling. A

simple rule of thumb for stability and accuracy for an allpass filter of an arbitrary order is

N − 0.5 ≤ △ < N + 0.5 [118]. Consequently, a desired delay length is best approximated

with an allpass filter of the order equal to the integer nearest to the ideal delay value.

Alternatively, a low-order allpass filter in cascade with a delay line can be used in order

to reduce the use of high-order filters, and hence limit computational load, as depicted

in Figure 3.11. However, such a simplified approach reduces the accuracy of the phase

delay at high frequencies compared with a higher-order allpass filter. In addition, there is

a danger of zero-pole cancellation when the first-order allpass filter is used to approximate

a zero delay, and hence even approaching zero delay value should be generally avoided

because of inevitable round-off errors. Consequently, it should be ensured that a delay

range lies totally above zero, i.e. 0.1 ≤ △ ≤ 1.1 [106].


3.6 Summary

A number of modelling techniques applicable to simulations of room acoustics were re-

viewed in this chapter, with the main focus on the equivalence of various approaches.

Geometrical and wave-based methods were briefly discussed and the motivation for the

chosen method was presented. In particular, an extensive overview of techniques that

are considered a subclass of FDTD methods was provided, which includes the digital

waveguide mesh, staggered and nonstaggered FDTD methods. The results of a consistent

approach to the analysis of the dispersion error and stability of these techniques clearly

indicates the equivalence of these three approaches. An important remaining question is

to find and analyse special cases of the general family of compact schemes, indicating the

most accurate and computationally efficient ones that are applicable to on-line simulations

of room acoustics.

For each technique, the boundary models available in the literature were briefly re-

viewed and the shortcomings of various boundary formulations were indicated. A strong

focus in subsequent chapters is to develop physically correct numerical formulations of

boundaries of the LRS type, which can be used for realistic and predictive FDTD simula-

tions.

68

Chapter 4

Compact FDTD Schemes

This chapter aims at providing a better insight into the 2D and 3D approximations of

the wave equation using compact FDTD schemes, which are applicable to 2D room and

membrane modelling, and 3D room acoustics simulations. In general, FDTD schemes can

be independently divided into a few categories. The first division concerns the number

of neighbouring grid points used in an update formula. Whereas compact schemes use

the nearest (neighbouring) grid points only in an update equation, large star systems ad-

ditionally take into account more distant nodes. The second division, which has already

been mentioned in the previous chapter, is related to the staggered or nonstaggered so-

lutions of the wave equation, for which the updates are conducted respectively midway

during each time step between two subgrids or once at each time step for the whole grid.

Finally, FDTD approximations can also be divided into explicit and implicit schemes.

While explicit methods calculate the state of the system at the next time step from known

previous and current discrete values, an implicit approach finds it by solving an equation

that additionally involves the state of the system at the next time step.

In this thesis, a family of compact FDTD schemes based on a rectilinear mesh is con-

sidered since large star systems (looking at more distant nodes in an update equation)

are inconvenient due to complicated treatment of boundaries. Such compact schemes are

restricted to three neighbouring nodes in a discrete space-time domain. The analysis pre-

sented in this chapter includes well known methods such as the standard leapfrog scheme

(that is mathematically equivalent to the digital waveguide mesh), the interpolated digital

waveguide mesh, the tetrahedral scheme in a 3D case, and compact implicit methods. An

overview of these techniques has been provided in Sections 3.2, 3.2.4, and 3.2.5. Compact

implicit schemes are not often used in an audio context, probably due to the extensive

cost of direct inversion of a multidimensional matrix at each time step. However, a fam-

ily of compact implicit schemes can be defined which enables efficient implementation

using alternating direction implicit (ADI) technique. This technique is characterised by

Chapter 4. Compact FDTD Schemes 69

an improved efficiency over the well established explicit schemes for simulations that re-

quire high accuracy, which makes compact implicit schemes an important option for audio

applications at high sample rates.

The presented special cases of the general family of compact schemes are analysed in

terms of stability, accuracy, dispersion error, and computational efficiency. The main focus

is on identifying schemes that are the most efficient for on-line modelling acoustic systems

in a specified audio bandwidth. Therefore, the smallest dispersion error and computational

efficiency are the main factors in the presented analysis. In addition, schemes that have

the most directionally independent dispersion error are analysed, and the most efficient

schemes are identified. The reader is reminded that we aim at on-line simulations of

acoustic spaces with moving sources and receivers, which excludes the use of off-line post-

processing techniques, such as frequency warping.

This chapter is divided in two parts, presenting the 2D and 3D compact schemes

approximating the wave equation, and each of these parts is structured as follows. Firstly,

the general formulation of a family of compact schemes is presented, followed by sections

identifying special cases of explicit and implicit schemes, respectively. Secondly, the issue

of stability and order of accuracy are investigated. Next, the dispersion error is evaluated,

followed by a detailed analysis of the accuracy and isotropy in valid frequency ranges for all

special cases of schemes. Finally, the computational efficiency is briefly analysed, followed

by some final conclusions.

4.1 2D Compact FDTD Schemes

The general objective of FDTD room acoustics modelling is the numerical solution of the

wave equation that governs sound wave propagation in air, which for a 2D x-y coordinate

system is given by∂2p

∂t2= c2

(∂2p

∂x2+

∂2p

∂y2

), (4.1)

where p denotes the acoustic pressure and c is the sound wave velocity. For rectilinear

meshes, a general equation for the family of compact implicit FD scheme approximating

the wave equation can be formulated as follows [42]

[1 + a(δ2x + δ2

y) + a2δ2xδ2

y ]δ2t p

nl,m = λ2[(δ2

x + δ2y)

+ bδ2xδ2

y ]pnl,m,

(4.2)

where λ = cT/X is the Courant number, and a and b are free numerical parameters. The

resulting scheme is implicit except for a = 0. The pressure update variable pnl,m, and


second-order derivative centered difference operators δ2t , δ2

x, and δ2y , are defined as

pnl,m ≡ p(x, y, t)

∣∣∣x=lX,y=mX,t=nT

, (4.3)

δ2t p

nl,m ≡ pn+1


l,m , (4.4)

δ2xpn

l,m ≡ pnl+1,m − 2pn

l,m + pnl−1,m, (4.5)

δ2yp

nl,m ≡ pn

l,m+1 − 2pnl,m + pn

l,m−1. (4.6)

4.1.1 Special Cases of Explicit Schemes

Setting a = 0 in Equation (4.2), a family of compact explicit schemes approximating the

2D wave equation results

δ2t pn

l,m = λ2[(δ2x + δ2

y) + bδ2xδ2

y ]pnl,m. (4.7)

Applying centered operators to Equation (4.7) yields the general compact explicit scheme

difference equation

pn+1l,m = d1(p

nl+1,m + pn

l−1,m + pnl,m+1 + pn

l,m−1)

+ d2(pnl+1,m+1 + pn

l+1,m−1 + pnl−1,m+1 + pn

l−1,m−1)

+ d3pnl,m − pn−1

l,m , (4.8)

where

d1 = λ2(1 − 2b), d2 = λ2b, and d3 = 2(1 − 2λ2 + 2λ2b). (4.9)

Choosing different values of parameter b yields special cases of compact explicit schemes,

the most important of which are provided in Table 4.1. The last two rows of the table

indicate the numerical cutoff frequencies in axial and diagonal directions, respectively.

The standard leapfrog (SLF) scheme results for b = 0, and we remind the reader here

that for the top value of the Courant number, this scheme is mathematically equivalent to

the rectilinear digital waveguide mesh (DWM), and also has the same numerical dispersion

and stability characteristics as Yee’s scheme. The stencil of the standard leapfrog scheme

is depicted in Figure 4.1, where only neighbouring nodes in axial directions are used in an

update formula.

As explained in [10], the rotated leapfrog (RLF) scheme can be interpreted as applying

standard centered finite operators along the diagonals rather than the horizontal and

vertical axes. One could say that the RLF stencil is 45o rotated in comparison to the SLF

scheme; both use a “6-point stencil” in the space-time grid to update the new value at

time (n + 1) (see Figure 4.1).

An interpolated stencil generally results with nonzero values of b; this can be viewed


Stanfard Leapfrog (SLF) Rotated Leapfrog (RLF)

Interpolated (INT) Implicit

(n+1)

(n)

(n-1)

(n+1)

(n)

(n-1)

(l-1,m-1) (l,m) (l+1,m-1)

(l+1,m)

(l+1,m+1)

(l-1,m-1) (l,m) (l+1,m-1)

(l+1,m)

(l+1,m+1)

(l-1,m-1) (l,m) (l+1,m-1)

(l+1,m)

(l+1,m+1)

(l-1,m-1) (l,m) (l+1,m-1)

(l+1,m)

(l+1,m+1)

Figure 4.1: Compact FDTD Stencils. For each type of stencil, the middle node at time (n + 1) isupdated using the remaining black-coloured nodes.

as a superposition of two leapfrog schemes, one of which is rotated by 45 degrees [11, 122].

Such schemes, for which a = 0 and 0 < b < 12 , have been referred to as ‘interpolated

schemes’ [10] and their mesh topology is illustrated in Figure 4.1. A nearly frequency-

independent dispersion error can be achieved by setting b = 16 ≈ 0.167, which yields

the nearly isotropic scheme described by Trefethen [115], which is referred here as the

interpolated isotropic (IISO) scheme. The interpolated digital waveguide mesh (IDWM)

developed by Savioja and Valimaki uses the same stencil, but the coefficients of their

difference equation are calculated using an optimisation method [93]; the result of which

is effectively equivalent to setting b = 0.1879 and λ = 1√2

in Eq. (4.8) [58]. Therefore,

it can be said that the IDWM as defined in [92] is nearly equivalent to the interpolated

isotropic scheme with the difference that the Courant number is substantially below its

top value.

Another interesting case is the interpolated wideband (IWB) scheme which results for

b = 14 and λ = 1. As will be explained later in this chapter, this is the only scheme in

the family that covers the whole available bandwidth (i.e. the lowest cutoff frequency is

positioned at the Nyquist frequency).

Note that the SLF, RLF, IISO, and IWB schemes use difference equation coefficients

with sizes that are powers of 12 , which makes these schemes particularly attractive for

efficient fixed-point implementations since all multiplications can be computed as bit shift


Table 4.1: Special cases of 2D compact explicit schemes, where faT and fdT denote the cutofffrequencies in axial and diagonal directions, respectively.

Standard Rotated Interp. Interp. Interp. Interp.Leapfrog Leapfrog DWM Isotropic Isotropic Wideband(SLF) (RLF) (IDWM) (IISO) λ2 = 0.5 (IWB)

a 0 0 0 0 0 0

b 0 12 0.1879 1

616

14

λ√

12 1

√12

√34

√12 1

d112 0 0.3103 1

213

12

d2 0 12 0.0949 1

813

14

d3 0 0 0.3794 −12

13 −1

faT 0.25 0.5 0.25 0.33 0.25 0.5

fdT 0.5 0.25 0.29 0.5 0.3041 0.5

operations. Conversely, the update formula for the IDWM has more complicated coeffi-

cients and hence it is not particularly well suited to fixed-point implementations.

4.1.2 Special cases of implicit schemes

As has already been discussed in Section 3.2.5, multidimensional compact implicit methods

approximating the wave equation require solving a difference equation at the advanced

time level. To address this issue, alternating direction implicit (ADI) methods have been

introduced [30], which rely on solving a set of three-diagonal equations. The problem

of inverting a matrix is then reduced to a succession of many one-dimensional problems

by factorising the scheme [19]. As presented in Section 3.2.5, a computationally efficient

splitting is

(1 + aδ2x)pn+1∗

l,m =λ2

a[−1 + (a − b)δ2

y ]pnl,m, (4.10)

(1 + aδ2y)δ

2t p

nl,m = pn+1∗

l,m +λ2

a(1 + bδ2

y)pnl,m, (4.11)


where pnl,m is the update variable and pn+1∗

l,m is an intermediate value. The implementation

of an ADI method in a 2D case consists of two following stages. Firstly, the intermediate

values are computed row by row in horizontal direction according to Equation (4.10).

Next, the intermediate values are used in computations in vertical direction according

to Equation (4.11). In order to preserve the global accuracy, the intermediate boundary

values should be obtained explicitly from Equation (4.11), as has been shown is Section

3.2.5. Writing out the standard difference operators in Equations (4.10) and (4.11) yields

the full difference equations

a(pn+1∗l+1,m − cpn+1∗

l,m + pn+1∗l−1,m) = d(pn

l,m+1 − epnl,m + pn

l,m+1), (4.12)

a(pn+1l,m+1 − cpn+1

l,m + pn+1l,m−1) = pn+1∗

l,m + f(pnl,m+1 + pn

l,m−1)

+gpnl,m − a(pn−1

l+1,m − cpn−1l,m + pn−1

l−1,m), (4.13)

where the respective coefficients are as follows

c = 2 − 1a ,

d = λ2(a−b)a ,

e = (2 + 1a−b),

f = 2a + λ2ba ,

g = 2 − 4a + λ2

a (1 − 2b).

(4.14)

For fast implementation of matrix inversion, there exists an efficient algorithm to inverse

the three-diagonal system of linear equations that arise in implicit finite difference schemes,

namely, the Thomas algorithm [109]. This technique has been described in detail in Section

3.4. All of the presented implicit compact schemes use the same update stencil illustrated

in Figure 4.1, in which all the neighbouring nodes in time and space domain are applied

in an update formula.

An interesting set of parameters (a = 1−λ2

12 and b = 16) has been proposed by Fair-

weather and Mitchell in [35], as presented in Table 4.2, for which fourth-order accuracy

in both time and space domain is obtained [19]. In comparison, compact explicit schemes

can be at the most second-order accurate in both time and space domain. This scheme

defines a subgroup of schemes within the family of Equation (4.2) of fourth-order accuracy

(FOA), as will be explained later in this section.

The maximally flat isotropic (MFI) scheme [122] is the scheme for which the difference

between axial and diagonal relative phase velocity is maximally flat. Such a scheme is

particularly useful when the aim is to apply off-line pre- and post-warping techniques

in order to remove as much as possible any direction-independent numerical dispersion,

such as those presented in Section 3.3. Finally, the optimum implicit (OI) scheme is an


Table 4.2: Special cases of 2D compact implicit schemes, where faT and fdT denote the cutofffrequencies in axial and diagonal directions, respectively.

Maximally Fourth-order OptimumFlat Implicit Accurate Implicit

(MFI) (FOA) (OI)

a 14 − 1

2√

31−λ2

12 0.0492

b 16

16 0.228

λ 1√

3 − 1 0.77

faT 0.38 0.29 0.33

fdT 0.5 0.5 0.5

optimisation example, that is computationally most efficient under the criterion that the

relative phase velocity error should not exceed 1% [122].

4.1.3 Stability analysis

The stability analysis presented in this section for a general family of 2D compact implicit

FDTD schemes is analogous to the stability analysis of a 1D standard leapfrog scheme

presented in Section 3.2.1. Let us first consider a wave travelling in direction x′ relative

to the Cartesian coordinate system, cutting the x-axis at an angle θ,

p(x′, t) = p0 este−kx′

, (4.15)

where p0 denotes a real-valued amplitude and s = σ + jω denotes complex frequency, and

k is the continuous-time wavenumber. Since the propagation direction can be expressed

as x′ = x cos(θ) + y sin(θ), the continuous-time single-frequency plane wave solution can

be written as

p(x, y, t) = p0 este−kxxe−kyy, (4.16)

where wavenumbers in both directions are given respectively as kx = k cos θ and ky =

k sin θ. This solution in the discrete space-time domain reads

pnl,m = p0 esnT e−kxlXe−kymX , (4.17)


where kx and ky denote the directional numerical wavenumbers. The centered finite differ-

ence operators given by Equations (4.4), (4.5) and (4.6) can for such solutions be expressed

as

δ2t p

nl,m =

(z − 2 + z−1

)pn

l,m,

= −4 sin2(ωT/2) pnl,m, (4.18)

δ2xpn

l,m = −4 sin2(kxX/2) pnl,m, (4.19)

δ2yp

nl,m = −4 sin2(kyX/2) pn

l,m. (4.20)

where z = esT , which represents the classic relationship between the s- and the z-domain

found in DSP literature.

Classical von Neumann stability analysis seeks a stability bound on λ such that no

growing solutions of the numerical system exist [109, 112]. Such a necessary stability

condition is expressed as |z| ≤ 1. Substituting Equations (4.18), (4.19), and (4.20) into

Equation (4.2) yields

z + 2B(sx, sy) + z−1 = 0, (4.21)

where following [10] the new variables

sx = sin2(kxX/2), (4.22)

sy = sin2(kyX/2) (4.23)

are introduced, and the variable B(sx, sy) is given by

B(sx, sy) = 2λ2F (sx, sy) − 1, (4.24)

where

F (sx, sy) =sx + sy − 4bsxsy

1 − 4a(sx + sy) + 16a2sxsy. (4.25)

The amplification equation (4.21) implies that the moduli of its two solutions cannot exceed

the value of 1 for any pair of wavenumbers (kx, ky), and hence also for any combination

of (sx, sy). Furthermore, it is sufficient to consider only real-valued wavenumbers in the

range −π/X ≤ kx, ky ≤ π/X as functions sx and sy are periodic with π. Note that kx

and ky can also become complex-valued [96], but in that case only the real part has to

be taken into account with regard to stability analysis. Equation (4.21) is stable when

|z| ≤ 1, which is equivalent to the following condition

B2(sx, sy) ≤ 1, (4.26)


which in turn yields a condition of the Courant number

λ2 ≤ 1

Fmax(sx, sy). (4.27)

Since the values of sx, sy are always in the range of [0, 1], the maximum value of function

F can only result for one of its extrema, which results in the following two cases

Fmax = max

(1

1 − 4a,

2 − 4b

1 − 8a + 16a2

). (4.28)

Thus the stability bound on the Courant number for a family of compact implicit finite

difference time domain schemes reads

λ2 ≤ min

(1 − 4a,

1 − 8a + 16a2

2 − 4b

). (4.29)

The Courant number can only take positive values, and hence the following constraints

on the parameters a and b can be ultimately calculated from Equation (4.29)

a ≤ 1

4, b ≤ 1

2. (4.30)

The order of accuracy of finite difference schemes is typically calculated in the FDTD

literature by substituting Taylor expansions of finite difference operators and substituting

them into the numerical wave equation. Next, the numerical error is calculated, which

serves as a basis for further analysis of free parameter values which bring about cancellation

of higher order terms in the error equation. The fourth-order truncation of the Taylor

expansions of the finite difference operators about the point (x, y, t), given respectively by

Equations (4.4), (4.5), and (4.6), yields

δ2t = T 2 ∂2

∂t2+

T 4

12

∂4

∂t4+ O(T 4), (4.31)

δ2x = X2 ∂2

∂x2+

X4

12

∂4

∂x4+ O(X4), (4.32)

δ2y = X2 ∂2

∂y2+

X4

12

∂4

∂y4+ O(X4). (4.33)

Firstly, these equations are used to substitute for the respective difference operators in

Equation (4.2), and next, the necessary substitutions of derivatives according to the mod-

ified equation method [125, 10] are made, which yields the overall numerical error that is


expressed as

E(p, a, b, λ,X) = −c2

[(a +

λ2 − 1

12

)∂4p

∂x4

+

(b − 1

6

)∂4p

∂x2∂y2

]X2 + O(X4).

(4.34)

This equation clearly indicates that the only set of free parameters for which a compact

scheme can achieve fourth-order of accuracy defines the FOA scheme. Consequently, all

the other schemes are only second-order accurate.

4.1.4 Numerical dispersion

An unwanted side effect of using numerical finite difference time domain (FDTD) schemes

is that they introduce a direction-dependent dispersion error. As has been explained in

Section 3.2.2, this artefact causes high frequencies to travel at a lower (or higher) speed

than the continuous-time wave speed, and it is in general dependent on the propagation

direction. As a measure of dispersion error, the numerical phase velocity relative to the

correct velocity is often used, which is defined as the ratio of the effective numerical wave

speed c over the real wave speed c, the former given as

c = ω/k. (4.35)

The amplification equation (4.21) can be used for deriving the formulae for the relative

phase velocity, which are next used to evaluate the dispersion error of all the presented

2D compact schemes. Let us first rewrite Equation (4.21) as

z − 2 + z−1 = −4λ2F (sx, sy). (4.36)

The dispersion relation is obtained by substituting for the left-hand side of this equation

with Equation (4.18), which yields

sin2(ωT/2) = λ2F (sx, sy). (4.37)

This formula enables one to calculate the angular frequency for a given pair of (sx, sy) as

ω =2

Tarcsin

(λ√

F (sx, sy)

). (4.38)

Finally, the formula for the relative phase velocity of 2D compact schemes takes the


following form

v(kx, ky) =w

k c=

2arcsin(λ√

F (sx, sy))

λ√

(kxX)2 + (kyX)2. (4.39)

Note that such a formula allows one to obtain the dispersion relation for the pair of

(sx, sy) and directional numerical wavenumbers (kx, ky). Thus it provides a numerical

solution for the relative phase velocity. However, it may sometimes be advantageous

to have an analytic solution of Equation (4.39), e.g. in order to calculate the value of

the dispersion error at a certain frequency. Such analytic formulae can only be derived

assuming one propagation direction. The derivation of the relative phase velocity for axial

directions is presented here as an example.

When the wave travels in one of the four axial directions, the wavenumber ka takes

either of the following values{

k2x = k2

a, k2y = 0

}or{k2

x = 0, k2y = k2

a

}. Hence the function

F can be next rewritten as a function of the axial wavenumber

F (ka) =sin2(kaX/2)

1 − 4a sin2(kaX/2). (4.40)

Such a function is next combined with the dispersion relation (4.37), and the numerical

wavenumber in the axial direction is written explicitly as

ka(ω) =2

Xarcsin

√sin2(ωT/2)

λ2 + 4a sin2(ωT/2). (4.41)

Finally, the formula for the relative phase velocity in axial directions yields

va(ω) =ω

ka(ω) c=

(ωT/2)

λ arcsin

√sin2(ωT/2)

λ2+4a sin2(ωT/2)

. (4.42)

Similar analytic formulae can also be obtained for other propagation directions, but are

omitted here for brevity (see [122] for more details).

Equation (4.41) can also be used to calculate the numerical cutoff frequency for axial

directions. A general function f(x) = arcsin(x) becomes complex-valued when x > 1.

Hence the largest possible argument of f(x) for which it still remains real-valued is x = 1.

From Equation (4.41), this condition corresponds to

sin2(ωT/2)

λ2 + 4a sin2(ωT/2)= 1, (4.43)


which after solving for the frequency yields

faT =arcsin(λ)

π√

1 − 4a. (4.44)

This formula gives the value of the cutoff frequency in axial directions, and the respective

values are presented in Tables 4.1 and 4.2. Note the for frequencies above this ‘cut-off’

frequency, waves are evanescent, and resonances are not sustained [96]. Consequently, the

numerical simulation is only valid for frequencies below the respective cutoff frequencies,

which vary for each propagation direction.

4.1.5 Dispersion analysis

The relative phase velocity for the special cases is illustrated in Figure 4.3. The values are

calculated using Equation (4.39) for a given set of wavenumbers components. These plots

are very indicative in assessing how ‘isotropic’ a particular scheme is. The interpolated

isotropic scheme for both its top Courant number λ2 = 0.75 and at a lower value λ2 =

0.5 have similarly directionally independent dispersion error to the interpolated digital

waveguide mesh. Furthermore, the MFI scheme is shown to be the most isotropic of the

three. These schemes are in general far more ‘round’ than the other special cases.

One disadvantage of these plots is that they do not allow simply to determine the

numerical dispersion at a given frequency. This issue has been addressed in some studies

(such as [120, 39]), where a circle was drawn on plots of this type with the intention of

indicating the “highest normalised temporal frequency”. Unfortunately, the wavenum-

bers (referred to in [120, 39] as spatial frequencies) are not proportional to the temporal

frequencies. This can easily be verified when inspecting the dispersion relation.

As explained in [122], this set is easily translated to a set of frequencies and propagation

angles, which allows plotting the relative phase velocity in polar form, where the radial

axis represents normalised frequency. The plots are limited to real-valued wavenumbers;

hence the plots end at a frequency above which waves are evanescent, and the dispersion

error is not that relevant. In other words, the edge of each plot marks the numerical cutoff

frequency. The lowest cutoff frequency also defines the useful bandwidth for a particular

scheme.

Figure 4.3 indicates that the standard leapfrog scheme (and thus the rectilinear digital

waveguide mesh and Yee’s staggered scheme) suffers from a strongly direction-dependent

dispersion error and has its lowest cutoff frequency (at 0.25fs) in axial directions. Con-

versely, the rotated leapfrog scheme has no dispersion error in axial direction but the valid

frequency range is determined by the lowest cutoff frequency in diagonal directions, which

effectively reduces useful frequency range to 0.25fs. The IDWM and the IISO scheme

are both nearly isotropic, and have their lowest cutoff frequency in axial directions which


kx X

k y X

SLF

−π 0 π−π

0

π

kx X

k y X

RLF

−π 0 π−π

0

π

kx X

k y X

IWB

−π 0 π−π

0

π

kx X

k y X

IISO

−π 0 π−π

0

π

kx X

k y X

IISO [λ2 = 0.5]

−π 0 π−π

0

π

kx X

k y X

MFI

−π 0 π−π

0

π

kx X

k y X

FOA

−π 0 π−π

0

π

kx X

k y X

OI

−π 0 π−π

0

π

color bar

relative phase velocity0.7 0.8 0.9 1

Figure 4.2: Relative phase velocity as a function of x- and y-direction wavenumbers. Refer back toTables 4.1 and 4.2 for acronyms. The darkness in the plots indicates the relative phase velocityerror, where white indicates a zero error, and any error larger then or equal to 0.3 is representedwith black.

reduces their useful bandwidth to 0.25fs and 0.33fs, respectively. Figure 4.3 also confirms

that the IDWM is almost equivalent to the IISO if one sets λ = 1√2. The IWB scheme has

no dispersion error in axial directions, and as can be seen from Figure 4.3, for this scheme

the wavenumber remains real-valued for all frequencies up to Nyquist, which means that

the whole available bandwidth is used. Dispersion is the strongest in diagonal directions

for this scheme. Concerning implicit schemes, all these schemes have the lowest cutoff fre-


quencies in axial directions and they are characterised by the full bandwidth in diagonal

directions. The valid frequency range for the MFI scheme amounts to 0.38fs, the FOA has

the lowest cutoff at 0.29fs in axial directions, and the OI scheme provides a bandwidth

up to 0.33fs.

Figure 4.3 shows that the FOA and the OI schemes have the largest bandwidth at

which the numerical dispersion is relatively small. Hence the implicit schemes are shown

to be generally more accurate than explicit compact schemes. Concerning explicit schemes,

the IWB and the ISO schemes perform relatively well in that regard, providing the widest

overall bandwidth in which the scheme is quite accurate in all propagation directions.

4.1.6 Accuracy and Isotropy

Since the largest and the smallest numerical dispersion errors consistently occur in either

the axial or the diagonal direction, perhaps the most useful and informative way of inspect-

ing the dispersion error is to plot the relative phase velocity only for these two directions,

as shown in Figure 4.4. In order to investigate the accuracy of compact schemes, we define

the accuracy range as the frequency band in which the relative phase velocity error does

not exceed a specified value, also referred to here as the critical error, for all propagation

directions. For 2D modelling, we define the admissible error as 2% to regard a simulation

accurate. We could also define the scheme isotropy as the frequency band in which the

difference between the axial and diagonal direction is smaller than 2%, as illustrated in

Figure 4.5 for explicit schemes.

The SLF scheme (and thus also the DWM and Yee’s scheme) is considered accurate

up to 0.1fs. Similarly, the RLF scheme is considered accurate only up to 0.1fs. The

IDWM is also accurate up to 0.1fs, and is isotropic up to 0.23fs with a high relative

phase velocity error of 15% at 0.23fs, but almost no isotropy error up to 0.2fs. The IISO

scheme is accurate up to 0.18fs, and is isotropic up to 0.27fs with a 3 times smaller overall

dispersion error of 5.5% at 0.27fs. The IWB scheme is accurate and isotropic up to 0.22fs.

Note that whereas the dispersion error of SLF, IDWM and IISO schemes becomes very

high immediately after exceeding the aforementioned frequencies, the IWB scheme error

grows very slowly up to almost the Nyquist frequency. The 2% error results for the MFI

scheme at 0.16fs, and it is isotropic up to 0.34fs. The FOA scheme is characterised by

the smallest dispersion error in the lowest range of frequencies of all the compact schemes.

However, its accuracy range (i.e., dispersion error smaller than 2%) is limited to 0.22fs,

similarly to the IWB scheme. The OI scheme has the largest range of frequency in which

it is still accurate, which amounts to 0.29fs, and the numerical error is also directionally

independent within this frequency range. Note that the relative phase velocity of the OI

scheme exceeds the value of 1 for frequencies in this range, which is illustrated in Figure


−135o

−90o

−45o

0o

45o

90o

135o

−180o

SLF

−135o

−90o

−45o

0o

45o

90o

135o

−180o

RLF

−135o

−90o

−45o

0o

45o

90o

135o

−180o

IWB

−135o

−90o

−45o

0o

45o

90o

135o

−180o

IISO

−135o

−90o

−45o

0o

45o

90o

135o

−180o

IISO [λ2 = 0.5]

−135o

−90o

−45o

0o

45o

90o

135o

−180o

MFI

−135o

−90o

−45o

0o

45o

90o

135o

−180o

FOA

−135o

−90o

−45o

0o

45o

90o

135o

−180o

OI

color bar

relative phase velocity0.7 0.8 0.9 1

Figure 4.3: Relative phase velocity as a function of frequency (polar plot radius) and propagationangle (polar plot angle). In each plot, starting from the most inner circle, the dotted-line circlesindicate f = (1

8, 1

4, 3

8, 1

2)fs. For acronyms, refer back to Tables 4.1 and 4.2.

4.5. The MFI scheme is characterised by the widest frequency range in which it is perfectly

isotropic, which spreads up to 0.3fs.

It is worth realising what the implications of the dispersion error on the room impulse

response are. For that purpose, a numerical simulation example of a rectangular membrane

with clamped edges is presented to compare the performance of both the compact explicit

and implicit methods. The simulated membrane consists of 8x8 grid points with outer

nodes assigned a constant value of zero. The term membrane is used here instead of a 2D


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.7

0.75

0.8

0.85

0.9

0.95

1

1.05


Rel

ativ

e ph

ase

velo

city

AXIAL DIRECTION

SLFRLFIWBIISOMFIFOAOI

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.7

0.75

0.8

0.85

0.9

0.95

1

1.05


Rel

ativ

e ph

ase

velo

city

DIAGONAL DIRECTION

SLFRLFIWBIISOMFIFOAOI

Figure 4.4: Relative phase velocity of 2D compact schemes for axial (top) and diagonal directions(bottom).

room due to the simple boundary implementation, which is characteristic for modelling

membranes with clamped edges. A sharp impulse was injected into one corner node (2, 2)

and the receiver was located in the opposite corner (7, 7) of the membrane. The magnitude

of the impulse responses for the interpolated digital waveguide mesh and the fourth-order

accurate compact implicit FD scheme are compared to theoretical modal frequencies in

Figure 4.6(a) and (b), respectively. In general, the numerical simulation brings about

systematic shifts in modal frequencies which increase with frequency. For the interpolated

digital waveguide mesh simulation, these frequency shifts are particularly strong, and as

a result the impulse response is considerably compressed. In comparison, the resonance

frequencies resulting with the compact implicit FD scheme match well with analytical

values for a significantly wider frequency range.

The effect of dispersion is also clearly visible in the time domain. Figure 4.7 illustrates

subsequent soundfields in a 2D coupled room simulated using the interpolated digital


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.02

0.04

0.06

0.08

0.1

Normalised frequencyDiff

eren

ce b

etw

een

diag

onal

and

axi

al p

hase

vel

ocity ISOTROPY ERROR

SLFIDWMIISO (λ2=0.5)IISOIWBMFIFOA

Figure 4.5: The absolute value of the difference between the relative phase velocity in diagonal andaxial directions of 2D compact schemes.

waveguide mesh and the fourth-order accurate compact implicit scheme. A sharp Gaussian

impulse was injected into a one grid point and the propagation of the resulting spherical

wave with reflections from boundaries is shown. One immediate observation is that the

wavefront in the simulation using the interpolated digital waveguide mesh is not well

preserved. Small ripples that appear after the wavefront actually arise from the dispersion

as high frequencies of the injected signal travel slower than low frequencies. Consequently,

the wavefront is smeared out as the simulation proceeds. It can also be observed that the

dispersion error of the interpolated digital waveguide mesh is directionally-independent

as the same smearing is clearly visible for all propagation directions. In contrast, the

wavefront in simulations using the fourth-order accurate compact implicit scheme is much

better preserved, which is a consequence of substantially smaller dispersion than for the

IDWM.

4.1.7 Computational Cost

Even though the figures presented in the previous section are very useful for investigation

of the dispersion error for a given sample rate, the direct comparison of these schemes in

terms of their computational efficiency is difficult to determine. This is primarily due to

the use of different Courant numbers for most of these schemes. Even though some insight

can be gained from comparing schemes for equal Courant numbers (see, for example, [10]),

it seems more informative to restrict any comparisons to schemes at their highest possible

value of the Courant number (provided that λ ≤ 1), since that is when the numerical


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−40

−30

−20

−10

0

10

20

30

40

50


Impu

lse

resp

onse

mag

nitu

de [d

B]

(a) Interpolated digital waveguide mesh

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−40

−30

−20

−10

0

10

20

30

40

50


Impu

lse

resp

onse

mag

nitu

de [d

B]

(b) Fourth-order accurate compact implicit scheme

Figure 4.6: Magnitude of the membrane impulse response for: (a) the interpolated digital waveguidemesh without frequency warping, and (b) the fourth-order accurate compact implicit scheme. Dashedlines denote the ideal mode frequencies of the membrane.

dispersion is generally the lowest.

Let us first consider the overall number of arithmetic operations and memory access

operations for each junction. Disregarding the differences in the values of the Courant

number, one can subdivide all the compact schemes into three categories based on the

type of implementation, namely the standard leapfrog scheme, the interpolated schemes

and compact implicit schemes implemented using the ADI technique. This way the number

of arithmetic operations and memory access operations per junction at each time step for

the three types of schemes can be calculated. Results of such analysis are presented in

Table 4.3. The number of memory locations required for storing values from previous time

steps is the same for all three categories. In comparison to explicit schemes, the increase in

the number of multiplications for compact implicit scheme is offset by the lower number

of memory access operations per grid point. Such a small number of memory access

operations results for computationally efficient splitting in the ADI technique given by


IDWM (t = 15 ms) FOA (t = 15 ms)

IDWM (t = 62.5 ms) FOA (t = 62.5 ms)

IDWM (t = 125 ms) FOA (t = 125 ms)

IDWM (t = 250 ms) FOA (t = 250 ms)

Figure 4.7: Soundfields in a 2D coupled room using the interpolated digital waveguide mesh and thefourth-order accurate compact implicit scheme, at four consecutive points in time.


Table 4.3: Arithmetic operations and memory access operation count.

Standard Inter- Fourth-orderLeapfrog polated Accurate(SLF) (INT) (FOA)

Sum/Sub 5 9 7

Mult/Div 2 3 16

Mem Acc 6 10 7

Equations (4.10) and (4.11).

In order to obtain the same numerical error as the IWB scheme or ADI schemes up

to 0.2fs, the standard leapfrog scheme would require a two times higher sample rate,

which would result in a four times denser grid and two times the number of samples to be

computed. Consequently, the number of arithmetic operations would increase eight times

and four times more memory locations would be needed. Note that for the same sample

rate the number of arithmetic operations and memory access operations is half that of the

interpolated schemes.

Given an accuracy criterion presented in the previous section, a suitable metric for

computational complexity is the amount of computation required in order to obtain a

certain accuracy (for example 2%) over a specified bandwidth. This metric seems very

reasonable since in the audio context one usually requires accurate results over a given

frequency range. Using the amount of nodal updates per square meter per second as a

metric for the computational load, and not considering the option of compensating for the

dispersion error via frequency warping, it has been shown in [122] that the IWB scheme

is always the optimally efficient nonstaggered compact explicit scheme regardless of the

choice of critial error value and around 4 times more efficient than the SLF scheme. The

IWB scheme is also optimally efficient of all compact explicit schemes, regardless of the

choice of the value of a tolerable numerical error, i.e. if we define the accuracy as 0.5%,

still the most efficient compact explicit FDTD scheme is the IWB scheme. Among the

remaining explicit schemes, the IISO scheme is also considered efficient [122]. Concerning

implicit schemes, they all require the same computational complexity per nodal update for

a given ADI implementation. The FOA scheme is the most efficient option, in particular

when high accuracy is required. Similar efficiency can also be obtained with the OI

scheme. As far as the comparison of explicit and implicit schemes is concerned, the analysis

presented in [122] has shown that the ADI method is always the most computationally


efficient as long as the critical error (i.e., accuracy limit) is up to about 2%. Setting a

substantially higher value of the critical error (e.g. at 10%), the IWB scheme is the most

efficient of all schemes. More results on the relative efficiency of compact FDTD schemes

can be found in [122].

4.2 3D Compact FDTD Schemes

There exist numerous practical applications for the FDTD technique in the area of ar-

chitectural design of auditoria, churches, listening rooms, and concert halls. Since real

acoustic spaces are three-dimensional, the numerical solution of the 3D wave equation is

the ultimate goal in numerical simulations of room acoustics. In a 3D x-y-z coordinate

system, sound wave propagation in air is governed by

∂2p

∂t2= c2

(∂2p

∂x2+

∂2p

∂y2+

∂2p

∂z2

), (4.45)

and the 3D compact implicit FDTD scheme approximating the 3D wave equation in the

form that enables an alternating direction implicit implementation is expressed as

(1 + aδ2x)(1 + aδ2

y)(1 + aδ2z)δ

2t pn

l,m,i = λ2[(δ2x + δ2

y + δ2z)

+ b(δ2xδ2

y + δ2yδ

2z + δ2

xδ2z)

+ cδ2xδyδz ]p

nl,m,i,

(4.46)

where a, b, and c are free parameters, λ = cTX denotes the Courant number and l, m,

and i denote spatial indexes in x-, y-, and z-direction, respectively. The centered finite

difference operators are analogous to their 2D counterparts, the example operator in a

new z-direction is given by

δ2ypn

l,m,i ≡ pnl,m,i+1 − 2pn

l,m,i + pnl,m,i−1. (4.47)

4.2.1 Special Cases of 3D Explicit Schemes

Similarly to the 2D case, this section deals with the family of 3D compact FDTD schemes,

which include both explicit and implicit schemes. Let us first consider an family of explicit

schemes approximating the 3D wave equation, which is obtained by setting a = 0. Hence


the general form of a compact explicit scheme is given as

δ2t p

nl,m,i = λ2[(δ2

x + δ2y + δ2

z )

+ b(δ2xδ2

y + δ2yδ

2z + δ2

xδ2z)

+ cδ2xδyδz]p

nl,m,i,

(4.48)

with two free parameters b and c, and the Courant number λ. Therefore, there exists one

more parameter for 3D schemes that can be adjusted for minimising the dispersion error

in comparison to the 2D case. The general formula for updating a grid point in a 3D

FDTD simulation using compact explicit schemes is obtained by applying centered finite

difference operators to Equation (4.48), which yields

pn+1l,m,i = d1

(pn


l,m+1,i + pnl,m−1,i + pn

l,m,i+1 + pnl,m,i−1

)

+ d2

(pn

l+1,m+1,i + pnl+1,m−1,i + pn

l+1,m,i+1 + pnl+1,m,i−1

+ pnl,m+1,i+1 + pn

l,m+1,i−1 + pnl,m−1,i+1 + pn

l,m−1,i−1

+ pnl−1,m+1,i + pn

l−1,m−1,i + pnl−1,m,i+1 + pn

l−1,m,i−1

)

+ d3

(pn

l+1,m+1,i+1 + pnl+1,m−1,i+1 + pn

l+1,m+1,i−1 + pnl+1,m−1,i−1

+ pnl−1,m+1,i+1 + pn

l−1,m−1,i+1 + pnl−1,m+1,i−1 + pn

l−1,m−1,i−1

)

+ d4 pnl,m,i − pn−1

l,m,i, (4.49)

where

d1 = λ2(1 − 4b + 4c),

d2 = λ2(b − 2c),

d3 = λ2c,

d4 = 2(1 − 3λ2 + 6λ2b − 4cλ2) (4.50)

are the coefficients used in implementation. The choice of the free parameters b and c

determines special cases of explicit schemes based on a rectilinear grid, and a list of the

main ones is provided in Table 4.4. This table presents information about free parameters

values (a,b,c), the stability bound on the Courant number for a given scheme, and the

values of coefficients used in the implementation, d1, d2, d3, and d4. In addition, the cutoff

frequencies in three directions are provided, namely in axial faT , side-diagonal fsdT ,

and diagonal fdT directions, respectively. We will refer to the side-diagonal direction

when considering the diagonal of the side of a cube, and to the diagonal direction when


Table 4.4: Special cases of 3D compact explicit schemes, where faT , fsdT and fdT denote the cutofffrequencies in axial, side-diagonal and diagonal directions, respectively.

Standard Octa- Tetra- Interp. Interp. Isotropic Interp.Leapfrog hedral hedral DWM Isotropic 2 Wideband(SLF) (OCTA) (TETRA) (IDWM) (IISO) (IISO2) (IWB)

a 0 0 0 0 0 0 0

b 0 12

14 0.2034 1

616

14

c 0 14 0 0.0438 0 1

48116

λ√

13 1 1

√13

√34

√34 1

d113 0 0 0.1205 1

41548

14

d2 0 0 14 0.0386 1

8332

18

d3 0 14 0 0.0146 0 1

64116

d4 0 0 −1 0.6968 −1 −98 −3

2

faT 0.196 0.5 0.5 0.196 0.333 0.333 0.5

fsdT 0.304 0.25 0.5 0.216 0.5 0.5 0.5

fdT 0.5 0.25 0.333 0.225 0.377 0.5 0.5

considering the diagonal of a cube.

The 3D standard leapfrog scheme (SLF) results for b = 0 and c = 0. For the top value of

the Courant number, the 3D standard leapfrog scheme is again mathematically equivalent

to the 3D rectilinear digital waveguide mesh, and also has the same numerical dispersion

and stability characteristics as the 3D Yee’s scheme. Furthermore, the implementation of

the standard leapfrog scheme at the top Courant number value is identical to the digital

waveguide mesh implementation using Kirchhoff variables. The stencil of the 3D SLF

scheme, depicted in Figure 4.8, consists of neighbouring nodes in three axial directions

only. Note that the presented cube represents spatial neighbouring nodes from the previous

time step.

The second scheme uses eight neighbouring nodes in diagonal directions only, as illus-


(l-1,m-1,i-1)

(l-1,m-1,i)

( l-1,m-1,i+1)

(l+1,m+1,i-1)

( l+1,m,i-1)

( l+1,m-1,i-1)(l,m-1,i-1

Standard Leapfrog

Octahedral (OCTA) Interpolated (INT)

Tetrahedral (TETRA)

)

(l-1,m-1,i-1)

(l-1,m-1,i)

( l-1,m-1,i+1)

(l+1,m+1,i-1)

( l+1,m,i-1)

( l+1,m-1,i-1)(l,m-1,i-1)

(l-1,m-1,i-1)

(l-1,m-1,i)

( l-1,m-1,i+1)

(l+1,m+1,i-1)

( l+1,m,i-1)

( l+1,m-1,i-1)(l,m-1,i-1)

(l-1,m-1,i-1)

(l-1,m-1,i)

( l-1,m-1,i+1)

(l+1,m+1,i-1)

( l+1,m,i-1)

( l+1,m-1,i-1)(l,m-1,i-1)

Figure 4.8: Compact FDTD stencils for the 3D standard leapfrog, the octahedral, the tetrahedral,and the 3D interpolated schemes.

trated in Figure 4.8. Because of the eight-point stencil used in an update formula, this

scheme is referred to in the FDTD and DWM literature as octahedral (OCTA) scheme

[11, 12, 17]. The octahedral scheme results for b = 12 and c = 1

4 , and has an advantage

that the top value of the Courant number amounts to λ = 1. Hence, the propagation

speed over one mesh point in axial directions is one sample.

The tetrahedral scheme [33] results for b = 14 and c = 0, and its name stems from the

number of twelve spatial grid points used in an update formula. The stencil of the tetra-

hedral (TETRA) scheme, illustrated in Figure 4.8, consists of twelve side-diagonal grid

points which are located in the middle of the edges of a cube defined for three propaga-

tion directions. Similarly to the octahedral scheme, the top value of the Courant number

amounts to 1.

The 3D interpolated stencil results for other values of free parameters b and c. In

general, interpolated schemes can be viewed as a superposition of three schemes, namely

the standard leapfrog, the octahedral, and the tetrahedral schemes. Hence, for most cases

its implementation consists of using all twenty six nearest neighbouring nodes surround-

ing the updated node. By analogy to the 2D case, such schemes can be referred to as

‘interpolated schemes’ [11] and their mesh topology is depicted in Figure 4.8. A nearly

frequency-independent dispersion error can be achieved by setting b = 16 and c = 0, which


is referred here as the interpolated isotropic (IISO) scheme. Another isotropic, in terms of

the dispersion error, scheme is obtained by setting b = 16 and c = 1

48 , which is referred here

as the interpolated isotropic (IISO2) scheme. Note that in the former isotropic scheme,

not all twenty seven surrounding mesh points are used in an update formula. Nevertheless,

we still consider it to be a part of the family of interpolated schemes. The 3D version

of the interpolated digital waveguide mesh (IDWM) developed by Savioja and Valimaki

uses the same 27-point stencil, but the coefficients of their difference equation are cal-

culated using an optimisation method [95]. The coefficients of the IDWM obtained by

optimisation up to 0.25fs (that have been presented in [95]) are effectively equivalent to

setting b = 0.2034, c = 0.0438 and λ = 0.5773 in Equation (4.49). In contrast to the 2D

case, these coefficients do not simply relate to the coefficients of the 3D IISO and IISO2

schemes.

As will be shown later in this chapter, the only scheme that actually provides the whole

available bandwidth in all propagation directions is the interpolated wideband scheme

(IWB). Such a 3D interpolated wideband scheme results for b = 14 , c = 1

16 and λ = 1.

The only schemes that have difference equation coefficients with sizes that are powers

of 12 are the octahedral, the tetrahedral, and the interpolated isotropic schemes. Hence

these schemes are particularly attractive for efficient fixed-point implementations since

all multiplications can be computed as bit shift operations. A set of simple difference

equation coefficients also results for the standard leapfrog and the interpolated wideband

schemes, in which case the multiplications by small odd numbers are required, in addition

to multiplications by powers of 12 . These can easily be implemented in fixed-point arith-

metic by multiplying by the closest power of 12 and adding the remaining value that is also

a power of 12 [18]. On the other hand, the IDWM and the IISO2 scheme have coefficients

(provided in Table 4.4) which cannot be efficiently implemented in fixed-point arithmetic.

4.2.2 3D Compact Implicit Schemes

The alternating direction implicit method can also be applied in the 3D case, when Equa-

tion (4.46) is factorised in order to reduce the issue of 3D matrix inversion to a succession

of 1D matrix inversion problems [19]. The most efficient splitting formula for the 3D ADI

method has been proposed in [42]

(1 + aδ2x)pn+1∗

l,m,i =λ2

a[−1 + (a − b)(δ2

y + δ2z )]pn

l,m,i, (4.51)

(1 + aδ2y)δ

2t pn+1∗∗

l,m,i = pn+1∗l,m +

λ2

a(b − a)δ2

ypnl,m,i, (4.52)

(1 + aδ2z)δ

2t (pn+1

l,m,i − 2pnl,m,i + pn−1

l,m,i) = pn+1∗∗l,m,i +

λ2

a(1 + bδ2

z )pnl,m,i, (4.53)


where pn+1∗l,m,i and pn+1∗∗

l,m,i are intermediate pressure variables. This splitting formula requires

that c = ab and hence the number of implicit schemes’ free parameters are reduced to two.

The computational procedure is analogous to that presented in Section 4.1.2, but in this

case it consists of three following stages in which intermediate values are calculated row

by row in all three respective propagation directions. The resulting 1D matrix inversion is

efficiently implemented using the Thomas algorithm, which has been described in detail

in Section 3.4.

The most accurate of all compact schemes results for the following set of parameters

(a = 1−λ2

12 and b = 16 ), which has been proposed by Fairweather and Mitchell [35]. These

parameters are exactly the same as in the 2D case and the resulting scheme is also fourth-

order accurate. Therefore, the fourth-order accurate (FOA) scheme is used for comparison

of compact implicit and explicit schemes in sections to follow. Note that the Courant

number also remains unchanged, and hence the fourth-order accuracy is obtained for

λ =√

3 − 1.

4.2.3 Stability Analysis

A single-frequency plane-wave solution of the wave equation in the continuous space-time

domain is in general given by Equation (4.15). Consider such a wave travelling in a positive

x-direction of a 3D coordinate system x-y-z, and cutting the Cartesian x-axis at a pair of

angles (θ, φ), where θ and φ denote the azimuth and elevation angles, respectively. Using

the coordinate rotation x′ = x cos(θ) cos(φ) + y sin(θ) cos(φ) + z sin(φ), the wave solution

becomes

p(x, y, z, t) = p0 este−kxxe−kyye−kzx, (4.54)

where kx = k cos θ cos φ, ky = k sin θ cos φ, and kz = k sin φ. In the discrete space-time

domain, this equation is expressed as

pnl,m,i = p0 esnT e−kxlXe−kymXe−kziX , (4.55)

where the directional numerical wavenumbers are given respectively as kx = k cos θ cos φ,

ky = k sin θ cos φ, and kz = k. sin φ. Therefore, such directional wavenumbers are repre-

sented by the effective wavenumber k and a pair of propagation angles (θ, φ).

Substituting the 3D versions of finite difference operators given by Equations (4.18),

(4.19), and (4.20) into the 3D compact FDTD scheme equation (4.46), the following am-

plification equation results

z + 2B(sx, sy, sz) + z−1 = 0, (4.56)

where the new variables are defined as sx = sin2(kxX/2), sy = sin2(ktX/2), and sz =


sin2(kzX/2), respectively, and

B(sx, sy, sz) = 2λ2F (sx, sy, sz) − 1, (4.57)

where

F (sx, sy, sz) =(sx + sy + sz) − 4b(sxsy + sxsz + sysz) + 16csxsysz

1 − 4a(sx + sy + sz) + 16a2(sxsy + sxsz + sysz) − 64a3sxsysz. (4.58)

The moduli of its three solutions have to be smaller than or equal to unity for any com-

bination (sx, sy, sz) and thus any combination (kx, ky, kz). Considering only real-valued

wavenumbers in the range −π/X ≤ kx, ky, kz ≤ π/X, the stability condition |z| ≤ 1 leads

to the following condition

λ2 ≤ 1

Fmax(sx, sy, sz). (4.59)

Since sx, sy, sz ∈ [0, 1], the maximum values of Fmax are given as

Fmax = max

(1

1 − 4a,

2 − 4b

1 − 8a + 16a2,

3 − 12b + 16c

1 − 12a + 48a2 − 64a3

). (4.60)

Therefore, the stability condition for the 3D compact FDTD schemes is

λ2 ≤ min

(1 − 4a,

1 − 8a + 16a2

2 − 4b,1 − 12a + 48a2 − 64a3

3 − 12b + 16c

). (4.61)

Since the value of the Courant number has to be positive, from Equation (4.61), the

domain of the free parameters is reduced by the von Neumann analysis to

a ≤ 1

4, b ≤ 1

2, c ≥ 1

16(12b − 3). (4.62)

The analysis of the order of accuracy of 3D compact FDTD schemes consists in sub-

stituting Taylor expansions of finite difference operators and substituting them into the

numerical wave equation. Similarly to the 2D compact schemes, there is only one scheme

that is fourth-order accurate, namely the 3D FOA scheme. All the remaining schemes

presented in this chapter are second-order accurate. Note that the overall numerical error

of second-order accurate schemes may vary quite substantially among them.

4.2.4 Numerical Dispersion

The numerical dispersion relation is derived from the amplification equation (4.56) by first

rewriting it as

z − 2 + z−1 = −4λ2F (sx, sy, sz), (4.63)


and next substituting z − 2 + z−1 = −4 sin(ωT/2), which yields

sin2(ωT/2) = λ2F (sx, sy, sz). (4.64)

The frequency for (sx, sy, sz) is therefore given as

ω =2

Tarcsin

(λ√

F (sx, sy, sz)

), (4.65)

and hence the final formula for the 3D compact FDTD schemes takes the following form

v(kx, ky , kz) =ω

k c=

2arcsin(λ√

F (sx, sy))

λ√

(kxX)2 + (kyX)2 + (kzX)2. (4.66)

The numerical solutions for the relative phase velocity are calculated from this for-

mula for a set of directional wavenumbers (kx, ky , kz). An analytic formulae for specific

propagation directions as a function of frequency can be found using similar procedure

to that presented in Section 4.1.4. In fact, the formula for the relative phase velocity in

axial directions is the same as for the 2D case. Thus the axial relative phase velocity

and the axial cutoff frequency are respectively given by Equations (4.42) and (4.44). The

respective analytic formulae for other propagation directions can also be found in a similar

fashion.

4.2.5 Dispersion Analysis

The largest and the smallest numerical dispersion errors consistently occur in one of the

following directions: the axial, the side-diagonal, or the diagonal direction. Therefore,

the most useful and informative way of inspecting the dispersion error is to plot the

relative phase velocity only for these three directions. Figure 4.9 illustrates the relative

phase velocity as a function of frequency for all the special cases of 3D compact schemes

presented in this chapter. The investigation of the cutoff frequencies of these plots is

also very indicative in assessing how ‘isotropic’ a particular scheme is and what its useful

frequency range is.

Let us first consider the isotropy and the valid frequency range of these schemes, which

are determined by the lowest cutoff frequency. As can be seen from Figure 4.9, both

interpolated isotropic schemes have nearly directionally independent dispersion error. In

addition, the interpolated digital waveguide mesh is characterised by the dispersion error

that is independent of the propagation direction. Therefore, these three schemes can be

considered much more ‘round’ in dispersion error than the other special cases.

Figure 4.9 indicates that the 3D standard leapfrog scheme (and thus the 3D rectilinear


digital waveguide mesh and 3D Yee’s staggered scheme) suffers from a strongly direction-

dependent dispersion error. The lowest cutoff frequency is observed in axial directions

and it amounts to 0.196fs. A higher value of the lowest cutoff frequency results for the

OCTA scheme, which is observed in side-diagonal and diagonal directions, illustrated in

Figure 4.9 and given in Table 4.4. This effectively reduces the useful frequency range

of the octahedral scheme to 0.25fs. An even larger useful frequency range is obtained

using the TETRA scheme, for which the lowest cutoff frequency is 0.33fs. Even though

the cutoff frequency for other two extreme propagation directions (i.e., axial and side-

diagonal directions) falls at Nyquist, the valid frequency range for the tetrahedral scheme

is reduced to 0.33fs.

As for the directionally-independent schemes, the IDWM has the lowest cutoff fre-

quency in axial directions which coincides with the respective cutoff frequency of the SLF

scheme. Therefore, the valid frequency range is limited to 0.196fs. The highest cutoff

frequency for the IDWM, which is observed in diagonal direction, amounts to only 0.225fs

making this scheme the most narrow (in terms of the frequency bandwidth) of all the

presented schemes. The reader is reminded that the spectrum is enlarged to a wider valid

frequency range after the warping technique is applied. In fact, the presented coefficients

of the IDWM have been optimised for the after-warping cutoff frequency of 0.25fs [95].

The other two isotropic schemes are characterised by a much wider valid frequency

range (up to 0.33fs). Note that the difference between the isotropic schemes and the

IDWM in a 2D case was not as substantial as in a 3D case. On the other hand, the

difference between the IISO and IISO2 is very minor, and it can only be observed for

diagonal directions, in which the respective cutoff frequencies are given as 0.377fs and

0.5fs, as presented in Table 4.4. Hence the IISO and IISO2 prove to be very similar in

their performance, as illustrated in Figure 4.9.

The IWB scheme is characterised by no dispersion in axial propagation directions, and

the numerical dispersion error is the strongest in diagonal directions. Furthermore, the

IWB scheme uses the whole available bandwidth, up to the Nyquist frequency. Hence the

simulations using the IWB scheme are valid up to 0.5fs. The FOA scheme has the lowest

cutoff frequency in axial directions, which reduces its valid frequency range to 0.242fs.

For the other propagation directions, nearly the whole bandwidth is provided.

4.2.6 Accuracy and Isotropy

Analogous to the 2D case, the accuracy criterion is given as the minimum frequency

for which the relative phase velocity error reaches a specified critical error value in the

direction characterised by the strongest dispersion. In this section, we apply the same

accuracy criterion defining the admissible error as 2% to regard a 3D simulation accurate.


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.7

0.75

0.8

0.85

0.9

0.95

1

1.05


Rel

ativ

e ph

ase

velo

city

AXIAL DIRECTION

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.7

0.75

0.8

0.85

0.9

0.95

1

1.05


Rel

ativ

e ph

ase

velo

city

SIDE DIAGONAL DIRECTION

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.7

0.75

0.8

0.85

0.9

0.95

1

1.05


Rel

ativ

e ph

ase

velo

city

DIAGONAL DIRECTION

SLFIDWMIWBOCTATETRAIISOIISO2FOA



Figure 4.9: Relative phase velocity of compact 3D schemes for axial (top), side-diagonal (middle), anddiagonal directions (bottom), respectively. Note that in the top plot, the relative phase velocity of thefollowing groups (IWB,OCTA, TETRA), (IISO and IISO2), and (SLF,IDWM) overlap, respectively.


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.02

0.04

0.06

0.08

0.1


Diff

eren

ce b

etw

een

best

and

wor

st p

hase

vel

ocity ISOTROPY ERROR

IDWMISOISO2IWBFOA

Figure 4.10: The absolute value of the difference between the relative phase velocity in two extremedirections of 3D compact schemes.

A similar criterion can be applied to istropy, where the admissible isotropy error between

the two extreme propagation directions is limited to 2%, as illustrated in Figure 4.10.

Applying the above criterion, the SLF scheme (and thus also the DWM and Yee’s

scheme) are considered accurate for up to 0.075fs. The OCTA scheme is accurate for up

to 0.093fs, and for the same frequency range it is also considered isotropic. The TETRA

scheme has a wider frequency range of accuracy which spreads up to 0.175fs. Similarly to

the OCTA scheme, the TETRA scheme has no dispersion in axial directions, and hence the

frequency range for isotropy coincides with the accuracy range (i.e., it is up to 0.175fs for

the TETRA scheme). The largest numerical error is encountered in diagonal propagation

directions for the IDWM. Therefore, the accuracy bandwidth of the IDWM scheme (before

warping) is reduced to a frequency range up to 0.069fs. Note that the IDWM is perfectly

isotropic for frequencies up to 0.18fs, for which the dispersion error does not exceed 1%,

which allows to obtain accurate simulations up to 0.25fs when the frequency warping

technique is applied. Applying the 2%, the IDWM is considered isotropic up to 0.191.

The IISO and the IISO2 schemes have their accuracy frequency range that extends to

0.175fs, which is the same as for the TETRA scheme. However, the range of frequencies

for which the IISO and the IISO2 are considered isotropic reaches as far as 0.269fs and

0.269fs, respectively. The IWB scheme is accurate for up to 0.186fs, which is also its

isotropy range; whereas the FOA scheme is considered accurate for up to 0.211fs and

isotropic up to 0.214fs.

The results for the relative phase velocity illustrated in Figure 4.9 for three propagation

directions indicate that there exist numerous schemes that are more accurate than the SLF


scheme. Concerning the accuracy of 3D schemes, the IWB scheme has the widest valid

frequency bandwidth of all explicit schemes in which the simulation is considered accurate,

regardless of the admissible error. Interestingly, the IISO, the IISO2, and the TETRA

schemes are considered accurate in exactly the same bandwidth, which is wider than for

the OCTA scheme, and substantially wider than for the SLF scheme and the IDWM,

regardless of the admissible error. Among all schemes, the widest frequency range of

accuracy is obtained for the IWB and the FOA schemes. However, direct comparison of

the IWB with the FOA scheme shows that the choice of the most accurate scheme depends

on the error criterion. For instance, for an admissible error of 2%, the FOA scheme is

accurate for a wider frequency band. On the other hand,if we define an admissible error

as 4%, the IWB scheme is accuate for up to 0.256fs, whereas the FOA scheme is accurate

for up to 0.242fs. Thus the IWB scheme provides a wider frequency range than the FOA

scheme in latter case. As far as isotropy in concerned, the IISO and the IISO2 schemes

have been shown to be isotropic for a significantly wider frequency range than the IDWM.

The main advantaged of the IDWM is that it is perfectly directionally-independent up to

0.18fs, which makes it such a good candidate for frequency warping.

4.2.7 Computational Cost

The computational complexity of 3D compact schemes can be evaluated in a similar man-

ner as for the 2D schemes. Analogous to [122], we can define the computational density

as the number of nodal updates per cubic meter per second as ρnu = fs

(X)3, where fs is the

sampling frequency required to guarantee accurate simulation (i.e. in which the relative

dispersion error does not exceed ec in any propagation direction), and X denotes the grid

spacing calculated from the Courant number λ as X = cfsλ . The required sample rate is

calculated as fs = ωc

min(ωaT,ωsdT,ωdT ) for the minimum normalised frequency in axial, side-

diagonal, and diagonal propagation directions for which the accuracy is ensured. ωc can

be set arbitrarily and it has to be applied consistently for calculating the computational

density ρnu of all compared schemes. Finally, the relative efficiency is defined as

ε(a, b, c, ec) =ρnu(0, 0, 0, ec, ωc)

ρnu(a, b, c, ec, ωc)(4.67)

so that ε(a, b, c, ec) is independent of the choice of ωc and the resulting value is normalised

by the computational density of the SLF scheme. The results of such a computational anal-

ysis for all cases of 3D compact schemes at their top Courant numbers and the following

values of the critical error ec = 2%, 4%, 8% are presented in Table 4.5.

Since each of the special cases is based on the rectilinear grid, each scheme requires

storing pressure values from two previous time steps. In this consideration, we do not

take into account the differences in amount of arithmetic operations per nodal update,


which is particularly true for fixed-point implementations. Furthermore, note that for an

alternating direction implicit technique, additional intermediate values are also required.

With regard to explicit schemes, the results presented in Table 4.5 indicate that the

IWB and both isotropic schemes (i.e. IISO and IISO2) are shown to be the most efficient of

all explicit schemes, independent on the numerical error that is admissible in a simulation.

In particular, the IISO and the IISO2 schemes are the most efficient computationally

for a tight accuracy criterion (up to 4%) and the IWB scheme is computationally more

efficient than the isotropic schemes for higher values of the accuracy criterion (above 4%).

However, since the relative efficiency values presented in Table 4.5 are very close to each

other, it can be concluded that the IWB, the IISO, and the IISO2 schemes are rather

comparable in efficiency.

The simulation using the TETRA scheme is slightly less efficient than using the IWB

and both isotropic schemes when low values of the critical error are considered. For

instance, in order to obtain the same dispersion error in a given frequency range at ec = 2%,

the TETRA scheme would require upsampling by 1.06fs, which means that the resuorces

would have to be increased by around 26%. Consequently, it can be concluded that the

efficiency of the IWB, the IISO, the IISO2, and the TETRA schemes is significantly better

than for other explicit schemes. Much more resources are required for the IDWM, the

SLF and the OCTA schemes to cover the same bandwidth at a given accuracy criterion,

regardless of the accuracy criterion applied.

The comparison of the best explicit schemes and the FOA scheme shows that for low

values of the admissible numerical error the FOA scheme is more efficient computationally

than explicit schemes. For ec = 2%, the FOA scheme is roughly 4 times more efficient

than the best performing explicit schemes, as given in Table 4.5, and this ratio increases

for a tighter error criterion. Therefore, the compact implicit method is shown to clearly

outperform compact explicit schemes in terms of efficiency when high accuracy is required.

It is only for substantially higher values of the admissible error (above 10%) when the best

performing explicit schemes can be considered more efficient than the FOA scheme.

4.3 Conclusions

In this chapter, a detailed description of a general family of compact explicit and implicit

FDTD schemes that are useful for modelling 2D and 3D acoustic systems is provided. For

modelling the room interior, the family of compact schemes includes a number of special

cases of interest that are either overall-accurate or nearly-isotropic, and as such are good

candidates for on- and off-line auralisation applications, respectively.

It is logical in the context of digital audio applications to evaluate schemes in terms

of the computational efficiency that guarantees that the numerical error does not exceed


Table 4.5: Relative computational efficiency for special cases of 3D compact schemes.

Stand. Octa- Tetra- Interp. Interp. Interp. 4th-ord.Leapf. hedral hedral DWM Isotrop. Wideb. Accur.(SLF) (OCTA) (TETRA) (IDWM) (IISO) (IWB) (FOA)

(IISO2)

λ√

13 1 1

√13

√34 1

√3 − 1

ε(2%) 1 0.44 5.63 0.7 8.67 7.01 29.64

ε(4%) 1 0.44 4.65 0.73 7.14 7.07 14.38

ε(8%) 1 0.44 3.46 0.8 5.31 6.99 7.16

a predescribed value in a given frequency range. Applying such an accuracy criterion, the

implicit schemes have been shown to be generally more efficient that the majority of explicit

schemes, apart from the case when large dispersion errors are admissible. The compact

implicit FD method with optimum parameter choice allows remarkable reduction of the

dispersion error, which combined with implementation using the alternating direction

implicit technique, makes it an efficient method for highly accurate on-line applications.

As for the explicit schemes, a new scheme has been identified that provides the full

bandwidth of a numerical simulation in both 2D and 3D case, which is not obtained

with any other compact scheme. For equal sample rates, the novel interpolated wideband

scheme is also the most accurate of all explicit schemes, regardless of the error criterion. As

far as computational efficiency is concerned, the interpolated wideband and the isotropic

schemes have been shown to be significantly more efficient than other explicit schemes in

both 2D and 3D, with just small differences in efficiency depending on the dimensionality

and the accuracy criterion. Therefore, it can be concluded that the interpolated wideband

and both isotropic schemes are shown to be the most efficient, which makes them good

candidates for on-line room acoustics applications. Note that the interpolated wideband

scheme additionally has an advantage of displaying no dispersion error in axial directions,

which is beneficial for room acoustics modelling since the most pronounced room modes

arise from reflections between parallel walls. The only rival for accurate on-line applica-

tions is the fourth-order accurate implicit scheme, which is generally more efficient when

the admissible value of the maximum numerical error is relatively low. However, the

relative complexity of the implementation of the compact implicit scheme using an ADI

technique probably makes the interpolated wideband and the isotropic schemes the best


choice for FDTD simulations of complex-shape domains based on a rectilinear grid.

In general, one could say that if frequency warping is applied, one of the identified

isotropic schemes (i.e., IDWM, IISO or IISO2) could be used, where the final choice de-

pends on the criterion associated with the application. Furthermore, the IISO scheme

has the advantage that the main scheme for updating the interior nodes can be ren-

dered efficiently using a fixed-point implementation, as opposed to the interpolated digital

waveguide mesh.

Finally, it could be argued that the perceptual importance of accurate room mode

frequencies gradually decreases with frequency. Hence from a perceptual point of view, the

advantage of the full bandwidth of the interpolated wideband scheme might well outweigh

its disadvantage of a slightly quicker rise in isotropy error (than for the isotropic schemes),

possibly making the IWB generally the best choice for room auralisation, regardless of

whether the application is on-line or off-line.

103

Chapter 5

FDTD Formulation of Locally

Reacting Surfaces

Reflections from boundaries of an acoustic space play a pivotal role in room acoustics, and

therefore attention has been given to the problem of formulating numerical approximations

of boundaries. Generally, realistic boundaries are frequency-dependent, and hence ap-

propriate boundary formulations should implement a frequency-dependent, complex wall

impedance. Developing such accurate formulations is an essential ingredient in creating

realistic and predictive FDTD simulations. Strictly speaking, complete physical models

of boundaries should include the transmission of waves in the wall. However, simulation

results in previous studies [15] have suggested that in many practical cases there is no

significant difference if wave propagation in the wall is neglected. Therefore, it is assumed

that any room surfaces are locally reacting, i.e. the reflective properties of any point on

the wall are completely characterised by a locally defined impedance.

With regard to FDTD modelling of a locally reacting surface, there exists a simple

frequency-dependent boundary model formulated for a staggered Yee’s grid that can have

mass-like and spring-like properties, which has been described in Section 3.2.3. With re-

gard to unstaggered grids, the vast majority of boundary models have been developed for

terminating of the digital waveguide mesh. All boundary formulations of the locally react-

ing surface type that are available in the DWM literature are based on a 1D formulation,

where wave propagation is locally assumed to be one-dimensional at a boundary, hence

the model does not implement wave propagation along boundaries. An overview of such

boundary terminations is presented in Section 3.2.4.

In this chapter, we show that the 1D formulation leads to significant errors in phase

and amplitude for low impedances or high angles of incidence, and propose improved

numerical formulations of locally reacting surfaces, that can be applied to both FDTD

and DWM modelling of acoustic spaces. The approach taken avoids the unphysicality

Chapter 5. FDTD Formulation of Locally Reacting Surfaces 104

of the 1D formulation, by combining numerical versions of the locally reacting surface

(LRS) condition and the multidimensional (i.e. 2D or 3D) wave equation. Besides using a

nonstaggered grid, the formulation also differs from Botteldooren’s (see Section 3.2.3 for

details) in that it does not introduce a new stability bound, i.e. the simulation is stable

for any physically feasible boundary parameters. The formulations supplied here consider

both 2D and 3D systems, including the treatment of corners and boundary edges. In

addition, a numerical boundary analysis method is proposed that provides a useful tool

for analysis of the numerical reflectance.

All the boundary update equations presented here are explicit difference equations,

and they are intended for room acoustics simulations using the standard leapfrog scheme.

However, provided that an unstaggered grid is used, these can in principle be used to

compute boundary nodes in all types of FDTD and K-DWM simulations, including cases

where the room interior nodes are computed with an implicit scheme. When alternating

direction implicit (ADI) methods are applied, the use of explicit boundary formulations

is not merely a choice but a necessity, as the ADI method requires at each time step that

the boundary node values are known before solving for the interior nodes, as explained in

Section 3.2.5. Note that such a combination introduces a scheme discontinuity, and thus

the overall stability of the simulation cannot be guaranteed in such cases.

The chapter is structured as follows. Section 5.1 briefly review the finite difference time

domain method and the theory of locally reacting surfaces, respectively. The next two

sections discuss the 2D formulation of frequency-independent and frequency-dependent

boundaries, followed by Section 5.4 that summarises all 3D formulations. The numerical

boundary analysis is proposed in Section 5.5. The discussion of all results is presented

in Section 5.6, including comparisons between different approaches as well as between

experiment and theory. Finally some concluding remarks are presented in Section 5.7.

5.1 Locally Reacting Surfaces

Throughout this and chapters to follow, reflecting boundary surfaces are assumed to be

planar, even though negligible roughness of the surface is permitted as long as its dimen-

sions are much smaller than the shortest wavelength [64]. A reflective boundary can be

modelled as a locally reacting surface (LRS), where the normal component of the particle

velocity at the surface of the wall depends on the sound pressure in front of the wall ele-

ment and not on pressure in front of neighbouring elements [64]. This assumption holds

for boundaries that are unable to propagate vibrations in the direction parallel to the

boundary surface. If we consider a sound wave travelling in a positive x-direction, the


boundary impedance Zw relates the sound pressure to the flow normal to the wall by

p = Zwux, (5.1)

where p denotes pressure and ux is the velocity component that is normal to the surface

of the boundary. For waves travelling in negative x-direction, this changes to p = −Zwux.

While the wave equation is derived from the principles of the conservation of mass and

the conservation of momentum, only the latter one may be applied in isolation at the

boundary [64]. For a boundary normal to the x-direction the conservation of momentum

equation is∂p

∂x= −ρ

∂ux

∂t. (5.2)

Differentiating both sides of Equation (5.1) yields

∂p

∂t= Zw

∂ux

∂t, (5.3)

Next, substituting for ∂ux

∂t in Equation (5.2) with Equation (5.3) yields the boundary

condition for the right boundary in terms of pressure only

∂p

∂t= −c ξw

∂p

∂x, (5.4)

where ξw = Zw/ρc is the normalised wall impedance, also known as the ‘specific acoustic

impedance’, which completely characterises the boundary. For the left boundary, the flow

is orientated in the opposite direction hence the minus in Equation (5.4) should be omitted

in that case. The corresponding planar wave reflection coefficient, herein referred to as

the reflectance, is related to the specific acoustic impedance and the angle of incidence by

(see Section 2.3.3)

R(θ) =ξw cos θ − 1

ξw cos θ + 1. (5.5)

Equation (5.5) implies that for ξw = 1, the specific acoustic impedance equals the charac-

teristic impedance of air, and the boundary is completely absorbent at normal incidence,

but not at any other angle. This shows that the locally reacting surface model is a priori

not a good basis for deriving an ‘anechoic boundary condition’. In summary, the LRS

theory provides a simple basis for modelling specular wall reflections, but is less suited to

modelling more complex boundaries such as anechoic terminations.


(a)

l,m-1p

l-1,mp

l,mp

l,m+1p

l+1,mp

l,m-1p

l-1,mp

l,mp

l,m+1p

l+1,mp

l,m-1p

l-1,mp

l,mp

l,m+1p

l+1,mp

(b) (c)

Figure 5.1: The leapfrog stencil in a 2D rectilinear mesh at: (a) a boundary, (b) an outer corner, (c)an inner corner. Ghost-point nodes are indicated with white-coloured circles and the room interior isindicated by grey shading.

5.2 Frequency-independent Boundaries

Even though real acoustic boundaries are characterised by complex impedances, let us for

now consider the simpler case of frequency-independent boundaries in a 2D acoustic space.

From a physical point of view, it is logical to treat the discretisation of a locally reacting

surface model as a 2D problem. However, the literature also provides a 1D approach,

which is why both cases are discussed here.

5.2.1 2D Formulation

An FDTD boundary model of a locally reacting surface can be obtained by approximating

the first-order spatial and time derivatives in Equation (5.4) with centered finite difference

operators, i.e.∂p

∂x=

pnl+1,m − pn

l−1,m

2X+ O(X2) (5.6)

∂p

∂t=

pn+1l,m − pn−1

l,m

2T+ O(T 2) (5.7)

The resulting equation can be written as an expression for the point lying outside of the

modelled space, also referred to as a ‘ghost point’ [113], which for the right boundary [see

Figure 5.1(a)] yields

pnl+1,m = pn

l−1,m +1

λξw(pn−1

l,m − pn+1l,m ). (5.8)


Eliminating the ghost point in the 2D discretised wave equation (3.5) using Equation (5.8)

yields the following 2D boundary update equation for a right boundary

pn+1l,m =

[2(1 − 2λ2)pn

l,m + λ2(pnl,m+1 + pn

l,m−1)

+ 2λ2pnl−1,m + (

λ

ξw− 1)pn−1

l,m

]/(1 +

λ

ξw

). (5.9)

For both the boundary update equation and the general scheme for updating the room-

interior nodes, the upper stability bound is matched by setting λ = 1√2, which reduces

Equation (5.9) to

pn+1l,m = r

[2pn


l,m−1

]+ (1 − 4r)pn−1

l,m , (5.10)

where r = ξw√2+2ξw

. Note that Equation (5.10) can be efficiently computed in fixed-point

implementations using only one multiplier and two bit shift operations. Similar efficient

forms can also be found for all boundary update formulae presented in subsequent sections,

but will not be explicitly provided as such.

5.2.2 1D Formulation

For the purpose of comparing the 2D formulation given above to the 1D model commonly

applied in the digital waveguide mesh approach, we also derive the boundary update

equation that follows from making the (unphysical) assumption that waves at the boundary

locally travel in x-direction only. In that case the discretised 1D wave equation applies

instead of the 2D wave equation. The starting point for derivation of a 1D boundary

model of a locally reacting surface is the same as for the 2D case, i.e. the discretisation of

the boundary condition, which gives Equation (5.8). However, this time the ghost point

is eliminated in the 1D discretised wave equation (3.4), which yields an update formula

for the boundary node

pn+1l,m =

[2λ2pn

l−1,m + 2(1 − λ2)pnl,m

+(λ

ξw− 1)pn−1

l,m

]/(1 +

λ

ξw

). (5.11)

For notational consistence, we keep m-indexes at the variables. As discussed in Section

3.2.2, it is natural to choose λ = 1√2

since that value of the Courant number is the upper

stability bound for the 2D rectilinear FDTD scheme given by Equation (3.5). Alternatively,

one could consider that the 1D wave equation has a stability bound at λ = 1, and locally

define the Courant number as such. If we then use Equation (5.11) and substitute ξw =1+R(0o)1−R(0o) , where R(0o) is the normal-incidence wall reflectance, the following boundary


(a) 2D boundary (b) 1D boundary

Figure 5.2: 2D and 1D terminations of a 2D mesh structure. Boundary nodes are indicated withgrey-coloured circles.

update formula results

pn+1l,m = [1 + R(0o)] pn

l−1,m − R(0o) pn−1l,m . (5.12)

This boundary update formula is identical to the formula used in many previous stud-

ies (see for example [89, 77]), and was originally derived from a digital waveguide mesh

perspective in [90]. For a given sample rate and a constant sound speed, λ defines the

distances between grid points. Therefore, choosing λ = 1 at the boundary implies that

the distance from the boundary node to the nearest neighbouring node in the medium

is smaller than distances between nodes in the medium. On the other hand, by setting

λ = 1√2

the grid spacing is uniform, but this choice should result in an increase in nu-

merical error at the boundary as this value is much below the upper stability bound for

the 1D wave equation. As will be shown in Section 5.6, the 1D formulation leads to erro-

neous results for either choice of the Courant number, which is fundamentally due to the

unphysical assumption of 1D wave propagation at the boundary.

5.2.3 Corners

In the 1D formulation, the boundary itself is effectively not regarded as part of the medium.

Hence at outer corners [see Figure 5.1(b)] the wave equation applies in neither x- nor y-

direction. Therefore outer corner nodes are not updated (i.e. kept at zero) when using a

1D boundary formulation, as depicted in Figure 5.2(b). Simple experiments have shown

that this leads to spurious results when a wave hits a corner; for example when a plane

wave travelling in x-direction reflects from a corner, some wave energy is directed in the

y-direction.

No such problems arise with the 2D formulation. The derivation of the update equation

for a top-right outer corner now starts from the principle that two boundary conditions


have to be satisfied simultaneously at the boundary node, namely

∂p

∂t= −c ξx

∂p

∂x,

∂p

∂t= −c ξy

∂p

∂y, (5.13)

where ξx denotes the normalised impedance in x-direction and ξy is the normalised impedance

in y-direction, respectively. Next, substituting for both ghost points in the 2D discrete

wave equation (3.5) with discrete versions of Equations (5.13), yields

pn+1l,m =

[2λ2(pn

l−1,m + pnl,m−1) + 2(1 − 2λ2)pn

l,m

+ (λ

ξx+

λ

ξy− 1)pn−1

l,m

]/(1 +

λ

ξx+

λ

ξy

). (5.14)

Hence the corner node can now be updated, as shown in Figure 5.1(b). Other outer

corners, which for 2D include boundaries left, above, and below the room interior are

derived in the same manner, where differences in the final formulae only occur in the form

of changes in sign and index; all coefficients remain the same.

At inner corners, also referred to as re-entrant corners, neither of the two boundary

conditions apply. There are also no ghost points to eliminate, as shown in Figure 5.1(c).

Hence an inner corner node can be updated with the discretised 2D wave equation (3.5).

5.3 Frequency-dependent Boundaries

In general, real acoustic boundaries are frequency-dependent, which means that the re-

flected wave has both phase and amplitude that differ from those of an incident wave,

and such changes vary with frequency [64]. The 2D LRS model presented in Section 5.2

can be extended by replacing the real-valued wall impedance with a frequency-dependent,

complex-valued impedance. As a first approximation, one can distinguish two basic types

of such boundaries, namely boundaries that behave spring-like and those that behave

mass-like [15]. The former can be used to model thin absorbing layers stretched on the

much harder boundaries; seat, floor and wall coverings are good examples. The latter

boundary type models heavy porous layers such as curtains or light nonstiff walls. More

complex boundaries can be treated as a superposition of these two types [15].

As will be shown in Section 5.6, the 1D boundary formulation of frequency-independent

boundaries suffers from severe amplitude and phase errors. For this reason the 1D ap-

proach is not considered for frequency-dependent boundaries.


5.3.1 Boundary Formulation

For a spring-like boundary, the impedance is defined as Zw(s) = R+K/s, where R denotes

resistance, K is the spring constant, and s is the Laplace frequency variable. Mass-like

boundaries are defined by Zw(s) = R+Ms, where M denotes the mass per unit area. For

some realistic acoustic boundaries, such as may be the case for parquet, both behaviours

need to be modelled. Therefore a fairly general boundary model can be defined by the

impedance

Zw(s) = R + Ms +K

s(5.15)

Inserting this impedance into Equation (5.4) yields the boundary condition for a right

boundary∂p

∂t= −c

(R

ρc

∂p

∂x+

M

ρc

∂p2

∂x∂t+

K

ρc

∫ t

−∞

∂p

∂xdt

), (5.16)

Centered finite difference operators [Equations (5.6) and (5.7)] are applied to approximate

the first-order derivatives ∂p∂t and ∂p

∂x in Equation (5.16). Concerning a numerical integra-

tion method, we propose the use of a composite trapezoidal rule with subintervals equal to

time steps. Trapezoidal integration is mathematically equivalent to the bilinear transform

s = 2T

1−z−1

1+z−1 applied to y = x/s, which yields

yn = Txn + xn−1

2+ yn−1, (5.17)

and therefore the numerical integration can be written

K

ρc

∫ t

−∞pn

l,mdt =TK

2ρc

n∑

i=−∞

(pi

l,m + pi−1l,m

). (5.18)

Similarly to the integration method, the bilinear transform is applied to approximate the

time differentiation in the mixed term ∂2p∂x∂t . This can be mathematically formulated as an

application of the bilinear transform to y = xs, which results in

yn =2

T(xn − xn−1) − yn−1. (5.19)

This ‘digitisation’ of the function of the spring- and mass-terms can be seen as an acous-

tic analogy to wave digital filter theory, in which the bilinear transform is used to map

electrically analogous circuit elements such as inductors and capacitors into their wave

digital equivalents [11]. As such, the proposed method for integration and differentia-

tion has some of the associated advantages, which include the property of mapping stable

analogue systems to stable digital systems. Another important feature of the proposed

discretisation is that an implicit formulation is avoided. Applying these transformations


to Equation (5.16) yields

pn+1l,m − pn−1

l,m = λ[aR(pnl−1,m − pn

l+1,m) + aKSnK − aMSn

M ], (5.20)

where the parameters aR, aK , and aM are given in Table 5.1. The new variable SK is

introduced to store the result of summation ‘up to now’, which is given by

SnK = pn

l−1,m − pnl+1,m + pn−1

l−1,m − pn−1l+1,m + Sn−1

K , (5.21)

and the new variable SM is introduced for storage of the results of the formula

SnM = pn

l+1,m − pnl−1,m + pn−1

l−1,m − pn−1l+1,m − Sn−1

M . (5.22)

In order to obtain the formula for the ghost point pl+1,m at time step n, the new storage

variables SK and SM are next written explicitly in Equation (5.20), which after some basic

algebra results in the ghost point formula

pnl+1,m = pn

l−1,m +1

λa(pn−1

l,m − pn+1l,m ) +

aK − aM

a

· (pn−1l−1,m − pn−1

l+1,m) +aK

aSn−1

K +aM

aSn−1

M ,

(5.23)

where the new parameter a is given in Table 5.1. Note that the boundary condition is

still centered at time step n resulting in a second-order accurate approximation, and the

grouping of terms in Equation (5.23) results in a structurally stable formulation. The

update formula for the right boundary node is then obtained by substituting for the ghost

point pl+1,m in the discrete-domain 2D wave equation (3.5) with the boundary condition

(5.23), which yields

pn+1l,m =

[λ2(2pn


l,m−1) + 2(1 − 2λ2)pnl,m

+(λ

a− 1)pn−1

l,m +λ2(aK − aM )

a(pn−1

l−1,m − pn−1l+1,m)

+λ2aK

aSn−1

K +λ2aM

aSn−1

M

]/(1 +

λ

a

). (5.24)

The boundary formulation requires the update of the boundary node, the ghost point,

and the new variables SK and SM (in that order) at each time step according to the

aforementioned equations. This way the computed values SK and SM are only applied in

the next time step. Note that only one previous value needs to be stored for the ghost

point.


Table 5.1: Numerical boundary parameters for a complex impedance.

Parameter Description

aR = Rρc Specific resistance

aK = TK2ρc Specific spring constant times T/2

aM = 2MTρc Two times specific mass divided by T

a = aR + aK + aM Sum of the previous parameters

5.3.2 Corners

The 2D formulation of corners is accomplished similarly to the frequency-independent case.

That is, an inner corner is updated with the 2D wave equation (3.5). An outer corner

has two boundary conditions (in x- and y-direction), the discretised versions of which are

used for the elimination of the two ghost points in the discretised 2D wave equation. The

resulting update formula for the top-right outer corner node is

pn+1l,m =

[λ2(2pn

l−1,m + 2pnl,m−1 + gn−1

x + gn−1y )

+( λ

ax+

λ

ay− 1)pn−1

l,m

+2(1 − 2λ2)pnl,m

]/( λ

ax+

λ

ay+ 1), (5.25)

where the ghost points are computed as

pnl+1,m = pn

l−1,m +1

λax(pn−1

l,m − pn+1l,m ) + gn−1

x , (5.26)

pnl,m+1 = pn

l,m−1 +1

λay(pn−1

l,m − pn+1l,m ) + gn−1

y , (5.27)

with

gn−1x =

aKx − aMx

a(pn−1

l−1,m − pn−1l+1,m)

+aKx

aSn−1

Kx +aMx

aSn−1

Mx , (5.28)


gn−1y =

aKy − aMy

a(pn−1

l,m−1 − pn−1l,m+1)

+aKy

aSn−1

Ky +aMy

aSn−1

My , (5.29)

where the four variables SKx, SMx, SKy, and SMy are computed recursively, in the same

way as Equations (5.21) and (5.22).

5.4 Boundaries in 3D


In the 3D formulation, the discretised boundary condition given by Equation (5.16) is

combined with the 3D discretised wave equation. From there on, the derivation is anal-

ogous to the derivation of the 2D model in Section 5.3.1. In this case, the ghost point

formula reads

pnl+1,m,i = pn

l−1,m,i +1

λa(pn−1

l,m,i − pn+1l,m,i) +

aK − aM

a

· (pn−1l−1,m,i − pn−1

l+1,m,i) +aK

aSn−1

K +aM

aSn−1

M ,

(5.30)

and the variables SK and SM are defined as

SnK = pn

l−1,m,i − pnl+1,m,i + pn−1

l−1,m,i − pn−1l+1,m,i + Sn−1

K , (5.31)

SnM = pn

l+1,m,i − pnl−1,m,i + pn−1

l−1,m,i − pn−1l+1,m,i − Sn−1

M . (5.32)

Finally, elimination of the ghost point in the 3D discrete wave equation (3.6) using the

boundary condition given by Equation (5.30) yields the 3D boundary update formula

pn+1l,m,i =

[λ2(2pn

l−1,m,i + pnl,m+1,i + pn

l,m−1,i + pnl,m,i+1

+pnl,m,i−1) + 2(1 − 3λ2)pn

l,m,i + (λ

a− 1)pn−1

l,m,i

+λ2(aK − aM )

a(pn−1

l−1,m,i − pn−1l+1,m,i)

+λ2aK

aSn−1

K +λ2aM

aSn−1

M

]/(1 +

λ

a

). (5.33)

This 3D formulation requires computing the formulae for the boundary node, the ghost

point, variables SK and SM at each time step with Equations (5.33), (5.30), (5.31), and

(5.32), respectively. The frequency-independent version of the 3D boundary model is

obtained by simply setting aK = 0 and aM = 0. In that case only the boundary node


given by Equation (5.33) needs updating.

5.4.2 Corners

The formulation of an outer corner in a 3D system consists in applying a 3D wave equation

together with three boundary conditions simultaneously. In the 3D case, three ghost points

have to be eliminated in the discrete 3D wave equation (3.6) with formulae obtained from

each boundary condition separately, namely in x-, y- and z-direction. This yields the

following update formula

pn+1l,m,i =

[λ2(2pn

l−1,m,i + 2pnl,m−1,i + 2pn

l,m,i−1 + gn−1x

+gn−1y + gn−1

z ) + (λ

ax+

λ

ay+

λ

az− 1)pn−1

l,m,i

+2(1 − 3λ2)pnl,m,i

]/(1 +

λ

ax+

λ

ay+

λ

az

). (5.34)

At an x-y edge of a 3D space, two boundary conditions together with the wave equation

are applied, which results in

pn+1l,m,i =

[λ2(2pn

l−1,m,i + 2pnl,m−1,i + pn

l,m,i−1 + pnl,m,i+1

+ gn−1x + gn−1

y ) + (λ

ax+

λ

ay− 1)pn−1

l,m,i

+2(1 − 3λ2)pnl,m,i

]/(1 +

λ

ax+

λ

ay

), (5.35)

with the ghost points computed using the formulae

pnl+1,m,i = pn

l−1,m,i +1

λax(pn−1

l,m,i − pn+1l,m,i) + gn−1

x , (5.36)

pnl,m+1,i = pn

l,m−1,i +1

λay(pn−1


y , (5.37)

pnl,m,i+1 = pn

l,m,i−1 +1

λaz(pn−1


z , (5.38)

where gx, gy, and gz are computed in the form of Equations (5.28) and (5.29), and the

six variables SKx, SMx, SKy, SMy, SKz, and SMz are computed recursively, in the same

way as Equations (5.21) and (5.22). As is the case for 2D systems, inner corner nodes are

updated using the discretised wave equation (3.6).


5.5 Numerical Boundary Analysis

The effective reflectance of a numerical boundary (hereafter referred to as the “numerical

reflectance”) can be predicted in an exact manner, and can also be used to determine

whether FDTD simulations using the 2D/3D boundary formulations are stable. To derive

such analytical formulae, methods that are very similar to standard methods for the

analysis of stability and numerical dispersion in FDTD schemes (see e.g. [112]) or the

GKSO (Gustafsson-Kreiss-Sundstrom-Osher) method for boundary condition analysis in

the frequency domain [44] can be applied. We will refer to this procedure as “numerical

boundary analysis” (NBA).

5.5.1 2D boundary

Consider a boundary in an x-y plane that is parallel to the y-axis and located at x = 0.

An incident plane wave of frequency ω propagating at an angle of incidence θ towards the

boundary from the area x < 0 can be expressed as [64]

p(x′, t) = p0 ejωt e−jkx′

, (5.39)

where p0 is the wave amplitude, k = ω/c denotes the wavenumber, and x′ = (x cos θ +

y sin θ). For the reflected wave, the sign in x-direction is reversed and the pressure ampli-

tude is multiplied by the reflectance. As shown in Section 2.3.1, the total sound pressure

at the boundary is the sum of the incident and reflected sound pressure

p(x, y, t) = p0 ejωt e−jky sin θ (e−jkx cos θ + R ejkx cos θ). (5.40)

The equivalent pressure in the discrete domain can be written as

pnl,m = p0 ejωnT e−jkmX sin θ

(e−jklX cos θ + R ejklX cos θ

). (5.41)

where p0 is the incident wave amplitude, R denotes the numerical reflectance, and k is

the discrete-domain wavenumber that can be computed for any real frequency ω from the

dispersion relation for the 2D standard leapfrog scheme given by Equation (3.20).

On the basis that all pressure variables involved obey the difference equation associated

with the leapfrog rectilinear scheme, Equation (5.41) can also be used for deriving all of


the other discrete-domain pressure values in the boundary update equation, for example

pn+1l,m = ejωT pn

l,m = z pnl,m, (5.42)

pn−1l,m = e−jωT pn

l,m = z−1 pnl,m, (5.43)

pnl−1,m = p0 ejωnT e−jkmX sin θ

·(e−jk(l−1)X cos θ + R ejk(l−1)X cos θ

), (5.44)

pnl+1,m = p0 ejωnT e−jkmX sin θ

·(e−jk(l+1)X cos θ + R ejk(l+1)X cos θ

). (5.45)

Let us first consider the frequency-independent boundary model given by Equation (5.9).

After substitution of all pressure values, the next step is to set l = 0, which corresponds

to x = 0 at a boundary. Solving then for the numerical reflectance R gives

R(z, θ) = −{(

1 +λ

ξw

)z −

[2λ2C + λ2(D + D−1)

+2(1 − 2λ2)]

+(1 − λ

ξw

)z−1

}

/

{(1 +

λ

ξw

)z −

[2λ2C−1 + λ2(D + D−1)

+2(1 − 2λ2)]

+(1 − λ

ξw

)z−1

}, (5.46)

where C = ejkX cos θ and D = ejkX sin θ. Note that the substitution of discrete-domain

pressure values in the above procedure is valid only when the 2D (or 3D) discretised

wave equation is imposed on the boundary node, which means that it is connected to

adjacent nodes on the boundary. Hence this method is not suitable for the evaluation of a

1D boundary formulation in a 2D/3D context (this includes all existing DWM boundary

formulations).

For the 2D frequency-dependent boundary, the numerical boundary analysis is per-

formed analogous to the frequency-independent case. The variable SK can in this case be

written as

SnK − Sn−1

K = pnl−1,m − pn

l+1,m + pn−1l−1,m − pn−1

l+1,m. (5.47)

For a plane wave of frequency ω, we can write this as

SnK = (pn

l−1,m − pnl+1,m)

1 + z−1

1 − z−1. (5.48)


where z = ejωT . Similarly, the expression for SM is given by

SnM = (pn

l+1,m − pnl−1,m)

1 − z−1

1 + z−1, (5.49)

Inserting these values in Equation (5.24) and solving for the numerical reflectance R yields

the final NBA formula

R(z, θ) = −{(

1 +λ

a

)z −

[2λ2C + λ2(D + D−1)

+2(1 − 2λ2)]

+ z−1[(1 − λ

a) + (C−1 − C)

·(λ2aK

a

2

1 − z−1− λ2aM

a

2

1 + z−1

)]}

/

{(1 +

λ

a

)z −

[2λ2C−1 + λ2(D + D−1)

+2(1 − 2λ2)]

+ z−1[(1 − λ

a) + (C − C−1)

·(λ2aK

a

2

1 − z−1− λ2aM

a

2

1 + z−1

)]}. (5.50)

Note that Equation (5.50) follows directly from the numerical boundary analysis applied

to Equation (5.24). However, it can also be rewritten in the mathematically equivalent

simple form given by Equation (5.46), where the specific acoustic impedance is defined as

ξw(s) = (R + Ms + K/s)/ρc and represented in the z-domain as

ξw(z) =a + 2(aK − aM )z−1 + (a − 2aR)z−2

1 − z−2. (5.51)

5.5.2 Stability

The NBA formula can also be used to prove the stability of the proposed boundary models

by showing that the boundary represents a passive termination. Passivity of an FDTD

boundary formulation is sufficient but not necessary condition for numerical stability and

guarantees that all of the system’s internal modes are stable [106]. Furthermore, such

a passivity proof is similar to the standard procedure of proving the stability of FDTD

schemes with the use of Von Neumann analysis [112] or the GKSO method for checking

the stability of boundary conditions for finite difference schemes [44].

Firstly, the 2D NBA formula given by Equation (5.46) can be reformulated as

R(z, θ) = − Q − 2λ2 C

Q − 2λ2 C−1, (5.52)


where for any numerical wavenumber such that −π/X ≤ k ≤ π/X, the new variable Q

can be expressed as

Q=[2 cos(ωT ) − 2λ2 cos(kX sin θ) + 2(1 − 2λ2)

]

+j

[2λ

ξw(z)sin(ωT )

]. (5.53)

For any frequency, the digital impedance filter response is complex-valued, i.e. we may

write ξw(z) = aw + j bw. Its multiplicative reciprocal is

1

ξw(z)=

aw

a2w + b2

w

− jbw

a2w + b2

w

, (5.54)

which now defines the variable Q as

Q=[2 cos(ωT ) − 2λ2 cos(kX sin θ) + 2(1 − 2λ2)

+2λ bw

a2w + b2

w

sin(ωT )]+ j[ 2λ aw

a2w + b2

w

sin(ωT )]. (5.55)

The boundary model is passive when∣∣∣R(z, θ)

∣∣∣ ≤ 1, which guarantees that there are no

growing solutions of the system with respect to time. From Equation (5.52), this condition

can be written as ∣∣Q − 2λ2 C∣∣ ≤

∣∣Q − 2λ2 C−1∣∣ . (5.56)

Taking the square of both the left-hand and the right-hand side of Equation (5.56) yields

∣∣Q − 2λ2 C∣∣2={Re{Q} − 2λ2 cos

(kX cos θ

)}2

+{Im{Q} − 2λ2 sin

(kX cos θ

)}2, (5.57)

and

∣∣Q − 2λ2 C−1∣∣2={

Re{Q} − 2λ2 cos(kX cos θ

)}2

+{

Im{Q} + 2λ2 sin(kX cos θ

)}2, (5.58)

where Re{Q} and Im{Q} denote real and imaginary parts of Q, respectively. Inserting

Equations (5.57) and (5.58) into Equation (5.56) and applying further algebraic manipu-


lations yields the following condition

− 2λaw

a2w + b2

w

sin(ωT ) sin(kX cos θ

)

≤ 2λaw

a2w + b2

w

sin(ωT ) sin(kX cos θ

). (5.59)

Since sin(ωT ) ≥ 0 for frequencies up to Nyquist, λ ≥ 0, and sin(kX cos θ

)≥ 0 is satisfied

for all angles −π/2 < θ ≤ π/2, Equation (5.59) can be finally reduced to

aw ≥ 0. (5.60)

Thus the effective numerical reflectance is passive provided that the real part of the digital

wall impedance is nonnegative, which follows directly from the definition of a positive real

impedance. Since the medium itself is lossless (none of the FDTD schemes used cause

numerical attenuation), it follows that for any λ ≤√

12 the simulation as a whole is always

stable.

5.5.3 3D Boundary

In a rectangular 3D coordinate system x-y-z with a wall normal to this system and located

at x = 0, an incident wave propagating in a positive x-direction is given by Equation (5.39)

with x′ = (x cos θ cos φ+y sin θ cos φ+z sinφ), where θ and φ are the azimuth and elevation

angles, respectively. The total sound pressure in the standing wave in the plane of the

boundary is given as

p=p0 ejωt e−jky sin θ cos φ e−jkz sin φ

· (e−jkx cos θ cos φ + R ejkx cos θ cos φ), (5.61)

where θ and φ are the azimuth and elevation angles, respectively, and the discrete wavenum-

ber k can be obtained from the dispersion relation for the 3D standard leapfrog scheme

given by Equation (3.21).

If we next write out the formulae in a similar manner to the derivation of expressions


in Section 5.5.1, the NBA formula for Equation (5.33) results

R(z, θ, φ) = −{(

1 +λ

a

)z −

[2λ2C + λ2(D + D−1)

+λ2(E + E−1) + 2(1 − 3λ2)]

+ z−1[(1 − λ

a)

+(λ2aK

a

2

1 − z−1− λ2aM

a

2

1 + z−1

· (C−1 − C))]}

/

{(1 +

λ

a

)z

−[2λ2C−1 + λ2(D + D−1) + λ2(E + E−1)

+2(1 − 3λ2)]

+ z−1[(1 − λ

a) + (C − C−1)

·(λ2aK

a

2

1 − z−1− λ2aM

a

2

1 + z−1

)]}, (5.62)

where C = ejkX cos θ cos φ, D = ejkX sin θ cos φ, and E = ejkX sinφ. In this case the wavenum-

ber k has to be computed from Equation (3.21). As for the 2D case, the NBA reflectance

magnitude can be shown to be less or equal to unity for all wavenumbers and angles of

incidence.

5.6 Results

5.6.1 Numerical Experiments

In order to investigate the properties of numerical boundary formulations, a 2D simula-

tion procedure for determining the numerical reflectance was designed. In this procedure,

simulations are executed using a grid consisting of 1800×1400 nodes. For most 2D simu-

lations, a highly accurate compact implicit ADI scheme (with λ = 1/√

2 for grid spacing

consistency) was used, but in specific cases the standard leap-frog scheme was applied; the

ADI scheme generally gives better reflectance results due to lower numerical dispersion,

which helps in creating more plane wavefronts. Furthermore, the ADI scheme was applied

for comparison of the 2D and 1D boundary models as it requires shorter simulation time

needed for the whole wavelet to arrive at a receiver position than the standard leapfrog

scheme room interior implementation. In such a scenario, the NBA does not exactly pre-

dict the numerical reflectance, but any deviations occur only at very high frequencies. The

size of the modelled room and the simulation time (2000 samples at the sample rate of

4kHz) were selected in such a way that only reflections from the investigated boundary

can reach a receiver position. The simulation time was chosen sufficiently long so that the


Source

Receiver Image receiver

Source

Receiver

(a) (b)

Figure 5.3: Schematic depiction of the simulation setup used in the numerical experiments.

whole wavelet can reach a receiver. The mesh was initialised with a sharp impulse injected

into a grid point. The source position was chosen so that 1) a constant distance of 400 grid

points from the centre of the investigated wall was preserved, and 2) the incident waves

at the following angles of incidence θ = 0o, 15o, 30o, 45o, 60o, 75o, resulted. Note that by

placing the reflecting boundary far enough from the sound source, the curvature of the

wavefront arriving at a boundary may be neglected. The exception was for θ = 75o, where

in order to maintain a flatter wavefront, the size of the modelled space was increased to

2800×2100, the distance from the source was increased to 700 points, and the simulation

time was extended to 3500 samples. When the leapfrog scheme was used for the room

interior implementation, the setup was altered in order to deal with the stronger dispersion

error (which affects the shape of the wavefront and leads to longer tails in the obtained

signals). In that scenario, the domain consisted of 2800×2100 nodes, the distance from a

source to the centre of the boundary was reduced to 100 points, and the simulation time

extended to 3700 samples in order to capture the whole wavelet reaching a receiver after

over 12 times the wavefront arrival time.

Each reflectance test procedure consists of two simulations, the setup of which is pre-

sented in Figure 5.3. Firstly, the investigated boundary is located in the middle of the

room, and next the wall is removed and the size of the simulated space increased by a

factor of two to 1800×2800 and 2800×4200 nodes, respectively. In the first simulation,

the signal reflected from the wall xf is measured at a receiver point located at the same

distance from the centre of the investigated boundary as the source position, as illustrated

in Figure 5.3(a). This signal inevitably also contains the ‘direct sound’. In the second sim-

ulation [see Figure 5.3(b)], two signals are measured, namely at the same receiver position

(this gives the direct sound xd) and at the image position of the receiver; the latter being

the freefield wave signal xi which arrives after travelling the same distance as the reflected


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Amplitude

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Amplitude

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−1

−0.5

0

0.5

1

1.5

2x 10

−3

Time [samples]Amplitude

θ=0o

θ=45o

θ=75o

Figure 5.4: Reflected signal (solid line) plotted against the theoretical reference signal (dashed line) forthe following angles of incidence θ = 0o, 45o and 75o, and a specific wall impedance ξw = 9.

signal. After the simulations, the direct sound signal is subtracted from the reflected signal

obtained with the first simulation to give the isolated reflected signal xr = xf − xd.

In order to analyse phase aspects of numerical boundary models, a time domain com-

parison of the reflected signals from the numerical experiments with the theoretical reflec-

tion signals is presented in subsequent sections. For frequency-independent boundaries,

the theoretical time domain signal, which is used as a reference, is obtained by multiply-

ing the freefield signal xi with the theoretical reflectance given by Equation (5.5). For

frequency-dependent boundaries, the ideal time domain signal is obtained as a time do-

main convolution of the freefield signal xi and the signal obtained from the inverse Laplace

transform of the theoretical reflectance given by Equation (5.5). The choice of the Laplace

transform was made to avoid the characteristic ripples caused by the inverse Fourier trans-

forms. In addition, all signals for time-domain plotting were filtered using a 41-tap low

pass FIR filter with a normalised cutoff frequency of 0.15fs in order to mask the (small)

effects of truncating the dispersive tail.

The performance of boundary models is further illustrated through the comparison

of the numerical and theoretical reflectance amplitude in the frequency domain. The

experimentally determined numerical reflectance is calculated as the frequency-domain


deconvolution of xr and xi, while the theoretical reflectance given by Equation (5.5) is

used as a reference. Furthermore, all measured signals were windowed with the use of the

right half of the Hanning window before applying Fourier transforms, in order to reduce

signal truncation effects.

The values of parameters related to specific resistance, mass, and spring were primarily

chosen to illustrate the numerical performance of the boundary models presented in this

chapter; they are not based on experimental data.

To empirically confirm the stability of boundaries and corners, a set of very long

simulations of small enclosures was executed, using the standard leapfrog scheme for im-

plementation of the room interior. Each boundary of the rectangular/cubic acoustic space

consisted of 10 nodes only in order to bring about a maximum number of reflections dur-

ing the 20 seconds long simulation. No unstable growth was detected in any of these

experiments, which confirms the stability proof provided in Section 5.5.2.

5.6.2 2D Frequency-independent Boundary

The results for analysis in time and frequency domains were obtained using the simulation

setup described in Section 5.6.1 with the ADI implementation of the interior of the room;

whereas in subsection on NBA, a standard rectilinear scheme was applied for the room

interior.

Time Domain Analysis

The comparison of the reflected signals for the 2D frequency-independent model and the

1D models with two Courant numbers is illustrated in Figure 5.4. The 2D model matches

exactly the expected signal in both time and amplitude. Conversely, time domain analysis

of both 1D boundary models indicates that a phase shift problem is introduced. The plots

show that for either choice of λ there is only one incident angle for which the 1D boundary

yields a correct phase (θ = 45o for λ = 1 and θ = 0o for λ = 1√2). For all other incidences,

a systematic phase error results, while the amplitude generally exceeds that of the ideal

theoretical reference signal.

Reflectance Magnitude Analysis

Figure 5.5 shows the reflectance obtained from numerical experiments (solid lines) plotted

against the theoretical reflection (dotted lines). Concerning the 2D model, the theoretical

reflection factor is very well matched for up to half the Nyquist frequency (denoted in

all figures as 0.25fs) even for very low values of the reflection coefficient and high angles

of incidence. In particular, the numerical reflectance adheres exactly to the theoretical

value at low frequencies, which is a desired property in FD modelling. Conversely, both


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litu

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θ=0o

θ=15o

θ=30o

θ=45o

θ=60o

θ=75o

Figure 5.5: Numerical reflectance amplitude (solid lines) plotted against theoretical reflectance ampli-

tude (dotted lines) for the following specific acoustic impedances ξw = 11

9, 7

3, 17

3, 10000 and angles of

incidence θ = 0o, 15o, 30o, 45o, 60o, 75o.


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Figure 5.6: Numerical reflectance amplitude of a 2D boundary formulation for ξw = 3

2, 7

3, 4, 9, 10000 and

normal incidence θ = 0o. Dashed lines denote the reflectance (obtained from numerical measurementsusing the standard leapfrog scheme for implementation of the room interior), dotted lines denote thetheoretical reflectance, and grey solid lines denote numerical reflectance obtained from the numericalboundary analysis (NBA).

1D models do not adhere to the theoretical reflection factor for the majority of angles of

incidence and reflection factors. In general, both 1D models approximate the magnitude

incorrectly at high angles of incidence, e.g. the amplitude deviation at θ = 75o is enormous.

This misalignment is particularly severe for low values of the reflection coefficient in the 1D

model with λ = 1 (used in DWM), which has a general tendency to outstrip the expected

amplitude at all incidences, and not even approximates theory at and around ω = 0.

This means that with such a boundary, there is no scope for improvement in accuracy

by increasing the sample rate. Note that the jumps near ω = 0 in the experimentally

obtained reflectance - particularly visible for 1D models with highest impedance value -

are artefacts due to the wavefront not being perfectly flat.

Numerical Boundary Analysis

Figure 5.6 shows an almost perfect match for θ = 0o between the NBA and the exper-

iments. As will be shown for frequency-dependent boundaries, similarly good matches

result for all the other angles of incidence and impedance values. This important result

validates both methods as tools for analysing numerical boundary formulations. In par-

ticular it enables to predict the numerical reflectance for leapfrog rectilinear scheme room

interior implementations.

In order to visualise the discrepancy between theoretical and numerical reflectance in


Figure 5.7: Numerical reflectance error of a 2D boundary model for a specific wall impedance ξw = 6.

a more generalised manner, one can define the following numerical reflectance error

ǫ(θ, ω) =

√R(θ, ω)2 − R(θ, ω)2, (5.63)

which takes into account both amplitude and phase deviations, because ǫ(θ, ω) represents

the distance between the complex numbers R(θ, ω) and R(θ, ω). As illustrated in Figure

5.7, the numerical error is most severe at high frequencies for angles of incidence in nearly

axial directions, which coincides with the fact that the numerical dispersion of the 2D

rectilinear scheme is the strongest in axial directions. On the other hand, the horizontal

white band indicates very low numerical error at and near θ = 45o, which coincides with

the fact the leapfrog scheme is dispersionless in diagonal directions.

5.6.3 2D Frequency-dependent Boundaries

Time Domain Analysis

The phase analysis of frequency-dependent boundary model is limited to one plot, since

similar results were obtained for all other angles; for more results see [61]. As depicted

in Figure 5.8, the complex impedance boundary model preserves the phase and the am-

plitude is very well matched even at very high angles of incidence. Such a correct phase

characteristic of this model is essentially down to the use of symmetric approximations

only, namely centered difference operators and bilinear transforms which have excellent

phase properties. In addition, the result of the boundary model proposed by Botteldooren

in [15] is illustrated in Figure 5.9 for comparison.


1120 1130 1140 1150 1160 1170 1180 1190 1200−4

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−2

−1

0

1

2

3

4

5x 10

−3

Time [samples]

Amplitude

Figure 5.8: Reflected signal (solid line) compared to the theoretical reference signal (dashed line) forthe frequency-dependent formulation, where θ = 60o and aR = 9, aK = 0.5, and aM = 10.

1120 1130 1140 1150 1160 1170 1180 1190 1200 1210 1220−4

−3

−2

−1

0

1

2

3

4

5x 10

−3

Time [samples]

Amplitude

Figure 5.9: Reflected signal (solid line) plotted against the theoretical reference signal (dashed line) forthe 2D complex impedance boundary model proposed by Botteldooren in [15] at the angle of incidenceθ = 60o, where aR = 9, aK = 0.5, and aM = 10.

Reflectance Magnitude Analysis

The numerical reflectance amplitude for two different boundary conditions is illustrated in

Figures 5.10 and 5.11, and different angles of incidence. Each figure presents reflectance

results for two example impedance values; for more results see [61]. The theoretical re-

flectance is matched well in general for up to a quarter of the sample rate for various

complex impedances and all angles of incidence. As with frequency-independent bound-

aries, the numerical reflectance adheres perfectly at low frequencies.

The reader is reminded here that the LRS theory assumes plane waves, while in the

experiments a spherical wave is excited. Even though the curvature of the wavefront

becomes increasingly plane with travelling distance, minor discrepancies remain, partly

also due to the mesh-induced dispersion error (this is particularly so when the leapfrog


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itude

θ = 75o

Figure 5.10: Numerical reflectance of a 2D complex impedance boundary for the following val-

ues of parameters: aR = 3

2and aR = 4, aK = 0.5, aM = 10 and angles of incidence

θ = 0o, 15o, 30o, 45o, 60o, 75o. Dashed lines denote the reflectance obtained from numerical mea-surements, dotted lines denote the theoretical reflectance, and grey solid lines represent numericalreflectance obtained from NBA.

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θ = 75o

Figure 5.11: Numerical reflectance of a 2D complex impedance boundary for the following values of pa-

rameters: aR = 3

2and aR = 4, aK = 0, aM = 10 and angles of incidence θ = 0o, 15o, 30o, 45o, 60o, 75o.

Dashed lines denote the reflectance obtained from numerical measurements, dotted lines denote thetheoretical reflectance, and grey solid lines represent numerical reflectance obtained from NBA.

scheme is used). This results in small jumps near ω = 0, that are particularly visible

at very high angles of incidence. However it can be seen from the NBA plots that such

sudden outstrips near ω = 0 do in fact not occur for the proposed boundary models.


Figures 5.10 and 5.11 show a perfect match between the measured and predicted

reflectance for θ = 45o, which is the angle of incidence for which neither the rectilinear

scheme used in the NBA derivations nor the compact implicit scheme used for the room

interior in numerical tests exhibit any numerical error. There is also a good match for

all possible angles of incidence and impedance values at low frequencies, where almost

no numerical error occurs. The difference between the simulated and predicted numerical

reflectance at high frequencies is due only to using a different scheme for the room interior.

A closer match of the reflectance obtained from experiments than from NBA indicates that,

despite the scheme discontinuity, a combination of the ADI method with the proposed

boundary model actually leads to a numerical reflectance that is closer to theory than

with using the leapfrog scheme consistently.

In addition, we have tested the application of the backward Euler method instead of

the bilinear transform for the numerical integration and differentiation associated with

the spring and mass boundary parameters, respectively. However, using such asymmetric

approximations resulted in a less accurate reflectance phase and amplitude, even though

the reflectance was correct in its dependence on the angle of incidence.

For comparison, we also have tested the boundary model proposed by Botteldooren

in [15], now using Yee’s classical staggered grid formulation to implement the interior.

Despite the fact that the backward Euler method is used for both differentiation and

integration, Botteldooren’s model for the staggered grid performs almost equally well to

the model presented in here for the unstaggered grid, and yields results that are correct

both in phase and amplitude. A possible explanation could be that the linear interpolation

that is applied in his method compensates for the asymmetry of the backward Euler

method.

5.6.4 3D Boundary Model

The reflectance magnitude of the 3D frequency-dependent model obtained with NBA for

various impedances, azimuth angles, and a constant elevation angle φ = 60o is depicted in

Figure 5.12 up to the cut-off frequency in the axial direction (i.e. up to 0.196fs). These

results indicate that the proposed boundary model yields highly accurate approximation

of the expected reflectance even at high incidences, for both azimuth and elevation angles,

and low impedance values.

The numerical reflectance error according to Equation (5.63) is illustrated in Figure

5.13 for a specific acoustic impedance 6, a constant azimuth angle of incidence θ = 45o,

and varying elevation angle φ. As in the 2D case, the wave equation is dispersionless at

diagonal directions, which occurs for θ = 45o and φ = arcsin(1/√

3) ≈ 35.3o. As can

be seen from Figure 5.13, the numerical reflectance error also vanishes in this direction.


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ctio

n a

mp

litu

de

θ = 0o

0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1


Re

fle

ctio

n a

mp

litu

de

θ = 15o

0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1


Re

fle

ctio

n a

mp

litu

de

θ = 30o

0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1


Re

fle

ctio

n a

mp

litu

de

θ = 45o

0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1


Re

fle

ctio

n a

mp

litu

de

θ = 60o

0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1


Re

fle

ctio

n a

mp

litu

de

θ = 75o

Figure 5.12: Numerical reflectance amplitude of a 3D complex impedance boundary for the following

values of parameters: aR = 3

2and aR = 4, aK = 0.5, aM = 10, the elevation angle φ = 60o,

and azimuth angles θ = 0o, 15o, 30o, 45o, 60o, 75o. Numerical reflectance obtained from the analyticevaluation method (solid lines) is plotted against theoretical values (dashed lines) up to 0.196fs.

Figure 5.13: Numerical reflectance error of a 3D boundary model for a real specific wall impedanceξw = 6 and azimuth angle θ = 450.

Hence it can be concluded that the numerical error generally manifests itself in the same

way for the boundary and the interior.


5.7 Conclusions

In this chapter, we have proposed a new method for constructing numerical formulations

of locally reacting surfaces that can be applied to finite difference time domain modelling

of room acoustics. Novel formulations were presented for simple frequency-dependent

boundaries in which the wall is characterised by a complex impedance expression that

incorporates linear resistance, inertia, and restoring forces. A good match with theory

was demonstrated for all angles of incidence and boundary parameters. Generally, these

boundary models yield accurate results both in reflectance phase and amplitude for fre-

quencies up to near the axial cut-off frequency (0.25fs for 2D case and 0.196fs for 3D

systems). As such, the proposed formulations represent a significant improvement over the

1D boundary formulations used in digital waveguide mesh modelling. It has been shown

that for frequency-independent boundaries, the 1D formulation structurally exhibits a

phase error, which implies that in a simulation of an enclosed space, the boundaries are

effectively positioned closer to each other than intended. In addition, the reflectance mag-

nitude of 1D boundary models systematically exceeds the theoretical reflectance value,

particularly so for high angles of incidence and low impedance values. While the anal-

ysis has focused on 2D simulations, these inaccuracies are even more pronounced in 3D

simulations, due to larger numerical errors and the increased range of possible angles of

incidence. The advantages of the proposed 2D/3D approach over the 1D approach also

apply to frequency-dependent boundaries. That is, the boundary filter formulation using

KW pipes, which until now is the main frequency-dependent boundary formulation avail-

able for unstaggered grid FDTD and K-DWM simulations, is based on exactly the same

principles as its frequency-independent counterpart, and in fact reduces to that when the

boundary filter is set to a constant.

The work described in this chapter also represents an improvement to Botteldooren’s

method, in that the use of symmetric finite difference operators leads to a boundary formu-

lation that does not introduce an additional numerical stability condition. Hence, unlike

with Botteldooren’s formulation for the staggered grid, which uses asymmetrical opera-

tors, the model presented here for the standard leapfrog scheme allows for a completely

free choice of boundary parameters.

In addition, we have introduced a novel analytic method for exact prediction of the nu-

merical reflectance of 2D/3D boundary models. This numerical boundary analysis (NBA)

method was validated by a precise match with experimental results. As such, the NBA

provides a fast and precise tool for evaluating multidimensional boundary formulations,

removing the need for carrying out elaborate numerical experiments that are time consum-

ing, require enormous computer memory, and are prone to small artefacts due to violation

of the assumption of plane waves.


In addition, the 2D/3D boundary formulation allows a proper formulation of corner

and edge nodes. In the literature, this problem appears to have never been addressed in

detail in the context of DWM and FDTD methods applied to room acoustics.

The boundary models derived in this chapter are intended for FDTD room acoustic

simulations with the use of the standard leapfrog scheme. The consistency in scheme

applied for the room interior and at the boundary implies that analytic techniques can be

used to unambiguously predict the performance of both the medium and the boundary.

Furthermore, the stability of the whole simulation is guaranteed. The same applies to the

mathematically equivalent rectangular DWM implemented using Kirchhoff variables, with

which the presented boundary models can be combined.

An alternative approach is to combine the boundary for the standard leapfrog scheme

with other schemes based on the rectilinear mesh, e.g. the ADI scheme or the interpolated

rectilinear mesh. For instance, in this chapter we used the ADI scheme to obtain accurate

reflectance results. However, due to the scheme discontinuity at the boundary, the stability

of such a simulation cannot be guaranteed, hence such a combination applied to modelling

a complex-shaped acoustic enclosure is not recommended for general use.

133

Chapter 6

Modelling Frequency-Dependent

Boundaries as Digital Impedance

Filters

In the previous chapter, it has been shown that a physically correct FDTD model of LRS

boundaries is formulated by combining the boundary condition (expressed in terms of the

wall impedance) with the multidimensional (i.e. 2D or 3D) wave equation. In contrast

to the 1D approach, the 2D/3D wave equation is now satisfied at a boundary, resulting

in a numerical formulation that really behaves as a locally reacting surface. That is, the

reflected sound wave approximates the theoretical LRS reflectance well for any value of the

wall impedance and angle of incidence, in particular at low frequencies. In addition, both

phase and amplitude are well preserved, which is not the case for 1D boundary models.

However, a simple frequency-dependent boundary model in which the wall is charac-

terised by a complex impedance expression that incorporates linear resistance, inertia, and

restoring forces, can only be considered a first-order approximation to a real-life bound-

ary, which is generally frequency-dependent. Alternatively, such a simple boundary model

could be used to approximate the boundary impedance in narrow frequency ranges, thus

creating a need for running numerous simulations in order to cover the whole audio band-

width. The output signals from a few simulations of the same acoustic space would need

to be filtered with the use of bandpass filters, and next combined in a one room impulse

response. However, such an implementation significantly increases the overall computa-

tional cost. Therefore, a more general approach should allow formulation of the boundary

for any complex impedance that fully represents the properties of a wall in the whole audio

bandwidth.

In this chapter, we extend a simple frequency-dependent boundary model to formu-

lating generalised frequency-dependent boundaries, which can be applied to both FDTD

Chapter 6. Digital Impedance Filter boundary model 134

and K-DWM modelling of acoustic spaces, by representing the boundary impedance with

a digital IIR filter of arbitrary order. The main challenge herein is to avoid obtaining an

implicit scheme, as well as avoiding the use of low-order accuracy finite difference oper-

ators. This is achieved by reformulation of the digital impedance filter (DIF) boundary

model in terms of suitable intermediate variables that can be computed recursively. In

addition, we provide a formula for the analytic evaluation of the numerical reflectance,

herein referred to as the numerical boundary analysis (NBA), which has been introduced

in Section 5.5.

The chapter is organised as follows. In Section 6.1, a locally reacting surface boundary

condition is presented starting from a digital impedance filter. Section 6.2 presents the

numerical formulation of the 2D FDTD frequency-dependent boundary model of a locally

reacting surface, including the treatment of corners. Section 6.3 presents the 3D DIF

boundary model, including the treatment of corners and boundary edges. The NBA

formulae for the proposed boundary models are provided in Section 6.4. Results obtained

from numerical experiments and the NBA are analysed in the frequency domain in Section

6.5 and compared with the 1D boundary model that connects boundary reflectance filter

with the FDTD room interior implementation with the use of the KW-pipe. A final

discussion is presented in Section 6.6, followed by some concluding remarks in Section 6.7.

6.1 Digital Impedance Filter (DIF)

In general, real acoustic boundaries reflect waves in a frequency-dependent manner. That

is, the reflected wave has a phase and amplitude that differ from those of the incident wave,

and such changes diverge with frequency [64]. This phenomenon can be incorporated in

an FDTD model with locally reacting surfaces by representing the specific acoustic wall

impedance with a digital filter.

The impedance filter can be obtained in various ways, some of which will be described

in Section 6.5. One approach is to measure the wall reflectance at a certain angle of

incidence, e.g. normal to the wall. The averaged results of such measurements are usually

available in the form of absorption coefficient values (α) provided for each octave band,

which can be converted to octave band reflection coefficients by |R| =√

1 − α [64]. Note

that such reflection coefficients do not include phase information. However, there are

several other techniques that also incorporate the reflectance phase information (such

as those presented in Section 2.5). For both cases, a normal-incidence digital reflectance

filter R0(z) can then be designed to approximate the measured reflection data in all octave

bands. Provided that this normal-incidence reflectance filter represents a passive boundary

(i.e. |R0(z)| ≤ 1), the transfer function of the digital impedance filter approximating an

appropriately physical (i.e. positive real) impedance, can be calculated using the inverted


form of Equation (5.5) with θ = 0o, i.e.

ξw(z) =1 + R0(z)

1 − R0(z). (6.1)

The general specific impedance is thus expressed in this study as an IIR filter, where both

nominator and denominator are of the same order N equal to the highest order of the

reflectance filter

ξw(z) =b0 + b1z

−1 + b2z−2 + ... + bNz−N

a0 + a1z−1 + a2z−2 + ... + aNz−N. (6.2)

Note that an IIR impedance filter ξw(z) results for reflectance filters of both FIR and IIR

type, and that direct design of ξw(z) from measured or modelled impedance data ξw(ω)

can lead to unphysical representations of the boundary (i.e. an impedance filter that is

not positive real).

6.2 2D DIF Model

This section presents the FDTD formulation of the 2D DIF model for right and left

boundaries in a rectilinear grid, an analogous K-DWM formulation, and the treatment for

outer and inner corners.


The boundary condition for a right boundary can be discretised by approximating the

first-order derivatives in time and space domains of the continuous boundary condition

given by Equation (5.4) with centered operators, which yields

pn+1l,m − pn−1

l,m

2T= −c ξw

(pnl+1,m − pn

l−1,m

2X

). (6.3)

In order to derive the multidimensional FDTD formulation of a locally reacting surface,

the discretised boundary equation (Equation 6.3) has to be combined with the discre-

tised multidimensional wave equation, which for the 2D standard leapfrog scheme is given

by Equation (3.5). Since the aim is to incorporate a frequency-dependent digital wall

impedance ξw(z), Equation (6.3) is first transformed to the z-domain

Pl,m(z − z−1) = −λ ξw(z) (Pl+1,m − Pl−1,m), (6.4)

where Pl,m denotes the z-transform of the discrete time-domain pressure pnl,m. To enable

incorporation of the filter in the FD boundary condition, it is necessary to write the


impedance filter in the following way

ξw(z) =b0 + B(z)

a0 + A(z), (6.5)

where filter nominator and denominator are given as

B(z) =N∑

i=1

biz−i, (6.6)

A(z) =

N∑

i=1

aiz−i. (6.7)

Now substituting Equation (6.5) in the discretised boundary condition given by Equation

(6.4) yields

Pl,m(z − z−1) = −λb0 + B(z)

a0 + A(z)(Pl+1,m − Pl−1,m). (6.8)

In order to obtain a computable boundary formulation, this condition is rewritten in terms

of the intermediate value Y

Pl,m(z − z−1) = λ Y, (6.9)

where

Y =b0 + B(z)

a0 + A(z)(Pl−1,m − Pl+1,m). (6.10)

Equation (6.10) can be treated as an input-output relationship Y (z) = ξw(z) X(z) with

the transfer function given by the IIR impedance filter ξw(z) and the filter input given as

X = Pl−1,m −Pl+1,m. We may rewrite this explicitly in terms of the input pressure values

as follows

Y =1

a0

[(b0 + B(z)

) (Pl−1,m − Pl+1,m

)− A(z)Y

]. (6.11)

This explicit filter formula is then split into two parts, namely the current filter input

values and a new term G which is computed using past filter values only

Y =b0

a0(Pl−1,m − Pl+1,m) +

G

a0, (6.12)

where

G = B(z)(Pl−1,m − Pl+1,m) − A(z)Y. (6.13)

Next, substituting Equation (6.12) into Equation (6.9) yields

Pl,m(z − z−1) = λ

[b0

a0(Pl−1,m − Pl+1,m) +

G

a0

]. (6.14)


l,m-1p

l-1,mp

l,mp

l,m+1p

l+1,mp

(b) Left Boundary

l,m-1p

l-1,mp

l,mp

l,m+1p

l+1,mp

(a) Right Boundary

Figure 6.1: The leapfrog stencil in a rectilinear mesh at: (a) a right boundary, (b) a left boundary.Ghost-point nodes are indicated with white-coloured circles and the room interior is indicated by greyshading.

Such a splitting is necessary in order to separate the current filter values Pl−1,m and Pl+1,m

from the explicit filter equation, both of which also appear in the discrete wave equation

and hence are needed for updating the boundary node. Whereas the first value can be

computed explicitly from the discrete wave equation, the second can only be updated from

the boundary condition, since Pl+1,m is an extra node located beyond the boundary as

shown in Figure 6.1(a) and thus no other acoustic laws apply at this node. In addition, the

introduction of intermediate variables appears to be a strict necessity, since avoiding this

(by multiplying both sides of Equation (6.8) with a0 + A(z) and working out a different

set of update equations) leads to a system that is not structurally stable (i.e. instabilities

can arise from numerical round-off errors in that case). The ‘ghost point’, which is defined

as a mesh node that is lying outside of the modelled space [113] [see Figure 6.1(a)], can

now be written explicitly as

Pl+1,m = Pl−1,m +a0

λb0

(Pl,m(z−1 − z)

)+

G

b0. (6.15)

and its inverse z-transform reads

pnl+1,m = pn

l−1,m +a0

λb0(pn−1

l,m − pn+1l,m ) +

gn

b0. (6.16)

Similarly, the inverse z-transform of Equation (6.12) is

yn =1

a0

[b0 (pn

l−1,m − pnl+1,m) + gn

], (6.17)

where yn is the filter output value at time step n and intermediate value g is the inverse

z-transform of G

gn =N∑

i=1

[bi

(pn−i

l−1,m − pn−il+1,m

)− ai yn−i

]. (6.18)


Finally, the update formula for the boundary node pl,m is obtained by substituting for the

ghost point in the discretised 2D wave equation given by Equation (3.5) with Equation

(6.16). Hence the final update formula for a boundary node is

pn+1l,m =

[λ2(2pn


l,m−1)

+2(1 − 2λ2)pnl,m + (

λa0

b0− 1)pn−1

l,m

+λ2

b0gn]

/(1 +

λa0

b0

). (6.19)

Since the filter input is given as xn = pnl−1,m − pn

l+1,m, we can next define the filter input

equation as

xn =a0

λb0(pn+1

l,m − pn−1l,m ) − gn

b0. (6.20)

Hence, the explicit filter difference equation is updated with

yn =1

a0

(b0 xn + gn

), (6.21)

where the intermediate value g is

gn =

N∑

i=1

(bi xn−i − ai yn−i

). (6.22)

Note that such a boundary formulation is valid when b0 6= 0 and b0 6= 1. This is the case

for realistic walls, as the first condition would require −1 reflection and the second would

mean no immediate reflection, which for the vast majority of real walls is not the case.

However, if the discretisation of the boundary would somehow result in the violation of one

of the aforementioned conditions, there are usually options available within the numerical

design procedure to avoid them.

The digital impedance filter (DIF) boundary formulation requires updating the inter-

mediate value g from the past filter input and output values only, and calculating the

boundary node pl,m, and the filter input and output at time step n according to Equa-

tions (6.22), (6.19), (6.20), and (6.21), respectively. Note that updating these equations

in the proposed order avoids the necessity to update the ghost point pl+1,m (i.e., the point

located outside of the modelled space). One IIR filter is necessary for each boundary

node, with the input to the IIR filter at node (l,m) defined as 6.20, and the intermediate

value g can be obtained directly from the impedance filter implementation as indicated

with Equation (6.22) and illustrated in Figure 6.2. Since the order of the impedance filter

nominator and denominator are always equal, the filter is most efficiently implemented in


Z-1

Z-1

Z-1

x(n)

y(n)

g(n)

a1a2aN

b1b2bN b0

Figure 6.2: The digital impedance filter implementation in a canonical form, indicating additional outputfor an intermediate value g.

canonical form.

6.2.2 Other Rectilinear-grid Boundaries

The derivation of the formulae for the other boundaries in a rectilinear grid (i.e. boundaries

on the left, upper, and lower sides) proceeds in the same manner, but involves changing the

relevant indexes and changing the sign of the flow normal to the boundary where required,

as described in Section 5.1. For example, the final update formula for the boundary node

of a left boundary (see Figure 6.1(b)) is

pn+1l,m =

[λ2(2pn

l+1,m + pnl,m+1 + pn

l,m−1)

+2(1 − 2λ2)pnl,m + (

λa0

b0− 1)pn−1

l,m

+λ2

b0gn]

/(1 +

λa0

b0

), (6.23)

Since the ghost point pl−1,m is computed according to

pnl−1,m = pn

l+1,m +a0

λb0(pn−1

l,m − pn+1l,m ) +

gn

b0, (6.24)

filter input x at time step n is defined as

xn = pnl+1,m − pn

l−1,m =a0

λb0(pn+1


b0, (6.25)

and the explicit filter difference equation is updated according to Equation 6.21, where

intermediate value g is given by Equation 6.22.


6.2.3 K-DWM Implementation

It is well known that the standard leapfrog FDTD scheme with the Courant number set to

its highest possible value is mathematically equivalent to the rectangular digital waveguide

mesh model [49]. Hence the boundary models presented here are also directly applicable

to K-DWM simulations. For instance, the formula for a right boundary node of the 2D

rectangular K-DWM is obtained by setting λ = 1/√

2, which reduces Equation (6.19) to

pn+1l,m =r

[2pn


l,m−1 +gn

b0

]

+(1 − 4r)pn−1l,m , (6.26)

where r = b02b0+

√2a0

, and filter input equation is updated according to

xn =

√2a0

b0(pn+1

l,m − pn−1l,m ) +

gn

b0, (6.27)

and the remaining filter equations are again computed from Equations (6.21) and (6.22).

Note that setting λ at its top value is generally a good strategy for FDTD simulation of

rooms, since it results in the most efficient implementation of the room interior as well as

the boundary. The efficiency arises from the smallest numerical error for top values of the

Courant number, as has been explained in Section 3.2.2.

6.2.4 Corners

The treatment of corners in the FDTD modelling is crucial for the overall stability of the

whole simulation, and therefore appropriate equations have to be formulated with great

care. A starting point for the derivation of update formulae for corner nodes is to realise

that a corner is considered a part of the modelled space, and hence the multidimensional

wave equation has to be satisfied at the corner node. In this case, a top-right outer

corner (depicted in Figure 6.3(a)) has to satisfy the 2D standard leapfrog scheme equation,

and the two ghost points are eliminated for with the use of two independent boundary

conditions in x- and y-direction, namely

∂p

∂t= −c ξx(z)

∂p

∂x,

∂p

∂t= −c ξy(z)

∂p

∂y, (6.28)

where ξx(z) and ξy(z) denote digital impedance filters in x- and y-direction, respectively.

The explicit formulae for the ghost points in both directions are given as

pnl+1,m = pn

l−1,m +ax

λbx(pn−1

l,m − pn+1l,m ) +

gnx

bx, (6.29)


l,m-1p

l-1,mp

l,mp

l,m+1p

l+1,mp

(b) Inner corner

l,m-1p

l-1,mp

l,mp

l,m+1p

l+1,mp

(a) Outer corner

Figure 6.3: The leapfrog stencil in a 2D rectilinear mesh at: (a) an outer corner, (b) an inner corner.Ghost-point nodes are indicated with white-coloured circles and the room interior is indicated by greyshading.

pnl,m+1 = pn

l,m−1 +ay

λby(pn−1

l,m − pn+1l,m ) +

gny

by, (6.30)

where gx and gy are computed from their respective filter implementations, bx and by

are the b0 values, and ax and ay are the a0 values for the x- and y-direction boundary

condition, respectively.

Substituting for these ghost points in a 2D wave equation with the discrete versions

of these boundary conditions results in the following update formula for a top-right outer

corner node

pn+1l,m =

[λ2(2pn

l−1,m + 2pnl,m−1 +

gnx

bx+

gny

by)

+2(1 − 2λ2)pnl,m + (

λax

bx+

λay

by− 1)pn−1

l,m

]

/(1 +

λax

bx+

λay

by

), (6.31)

where two filter equations in x- and y-diections should be calculated from their respective

filter equations.

Regarding treatment of inner corners - in the FDTD literature also referred to as

re-entrant corners - neither of the two boundary conditions [Equation (5.13)] apply. Fur-

thermore, there are no ghost points to eliminate [see Figure 6.3(b)], and therefore the

discrete 2D wave equation given by Equation (3.5) can be used as an update formula for

an inner corner node.

6.3 3D DIF Model

The formulation of the 3D DIF model, including the treatment of corners and edges, is

accomplished in a similar manner to the 2D case presented in Section 6.2.



The derivation of the update formula for a right boundary again consists in combining

the discrete wave equation with the discrete version of the boundary condition given by

Equation (5.4). Thus from the generalised frequency-dependent boundary condition, we

arrive at the 3D form of Equation (6.20)

pnl+1,m,i = pn

l−1,m,i +a0

λb0(pn−1


gn

b0, (6.32)

where l, m, and i denote spatial indexes in x-, y-, and z-direction, respectively. The digital

impedance filter difference equations are again given by Equations (6.21) and (6.22), for

which filter input is now defined as

xn = pnl−1,m,i − pn

l+1,m,i =a0

λb0(pn+1

l,m,i − pn−1l,m,i) −

gn

b0. (6.33)

Concerning a discrete 3D wave equation, the standard leapfrog scheme is given by Equation

(3.6) [11]. Substituting for the ghost point in Equation (3.6) with Equation (6.32) yields

the update formula for the right boundary node

pn+1l,m,i=

[λ2(2pn


l,m−1,i + pnl,m,i+1

+pnl,m,i−1) + 2(1 − 3λ2)pn

l,m +λ2

b0gn

+(λa0

b0− 1)pn−1

l,m

]/(1 +

λa0

b0

). (6.34)

This 3D formulation requires computing the intermediate value g from the past filter

input and output values only, and calculating the boundary node pl,m,i, and the filter

input and output at time step n according to Equations (6.22), (6.32), (6.20), and (6.21),

respectively.

6.3.2 Corners and Boundary Edges

In order to obtain a formula for an outer corner of a 3D system, the discrete 3D wave

equation should be combined with three boundary conditions simultaneously in x-, y-,

and z-directions, respectively, by which all three ghost points are eliminated for, and the


following update formula results

pn+1l,m,i =

[2λ2(pn

l−1,m,i + pnl,m−1,i + pn

l,m,i−1)

+λ2(gnx

bx+

gny

by+

gnz

bz) + 2(1 − 3λ2)pn

l,m,i

+(λax

bx+

λay

by+

λaz

bz− 1)pn−1

l,m,i

]

/(1 +

λax

bx+

λay

by+

λaz

bz

). (6.35)

In a similar way, the nodes at an x-y outer edge of a 3D space are updated according to

the formula derived as a combination of Equation (3.6) with two boundary conditions,

namely in x- and y-directions, which yields

pn+1l,m,i =

[λ2(2pn

l−1,m,i + 2pnl,m−1,i + pn

l,m,i+1

+pnl,m,i−1) + λ2(

gnx

bx+

gny

by) + 2(1 − 3λ2)pn

l,m,i

+(λax

bx+

λay

by− 1)pn−1

l,m,i

]/(1 +

λax

bx+

λay

by

).

(6.36)

As for the implementation of inner corners and inner edges, the corner node can be com-

puted simply with the 3D standard leapfrog scheme equation given by Equation (3.6).


As explained in Section 5.5, the numerical boundary analysis (NBA) is an analytic method

for the prediction of the numerical boundary reflectance. In comparison to the numerical

experiments discussed in Section 6.5, this method has the advantage that results can be

computed much faster, and are free from numerical artefacts. In Section 5.6, both the

analytic and the numerical approach to determining the numerical reflectance have been

validated by finding a high level of agreement between the respective results.

In this section, we provide NBA formulae for exact prediction of the numerical re-

flectance of boundaries modelled using the DIF approach explained in Sections 6.2 and

6.3.

6.4.1 2D DIF Model

Considering a wall normal to the rectangular coordinate system in the x-y plane parallel

to the y-axis and located at x = 0, the total sound pressure in the plane of a locally


reacting surface can be derived by adding incident and reflected sound pressure values.

The discrete-domain pressure at the right boundary node is given by Equation (5.41),which

is readily transformed to the z-domain by esT = z and denoting the z-transform of pnl,m

with Pl,m. We may also use Equation (5.41) as a basis for substituting all the other

pressure variables in the boundary update equation. Again applying z-transforms, some

examples of these substitutions are

pnl,m→ Pl,m, (6.37)

pn+1l,m →z Pl,m, (6.38)

pn−1l,m →z−1 Pl,m, (6.39)

pnl−1,m→p0 zn e−jkmX sin θ (6.40)

·(e−jk(l−1)X cos θ + R ejk(l−1)X cos θ

).

The variable g in Equation (6.19) is substituted for with a reformulated form of Equation

(6.13)

G =[B(z) − A(z) ξw(z)

] (Pl−1,m − Pl+1,m

)

= Hw(z)(Pl−1,m − Pl+1,m

). (6.41)

The NBA formula for the proposed DIF boundary model is then obtained by setting l = 0

(which corresponds to x = 0 at a boundary) and solving for the numerical reflectance R,

which yields

R(z, θ) = −{(

1 +λa0

b0

)z −

[λ2(2C + D + D−1)

+2(1 − 2λ2)]

+λ2

b0(C−1 − C) Hw(z)

+(1 − λa0

b0)z−1

}/

{(1 +

λa0

b0

)z

−[λ2(2C−1 + D + D−1) + 2(1 − 2λ2)

]

+λ2

b0(C − C−1) Hw(z) + (1 − λa0

b0)z−1

},

(6.42)

where C = ejkX cos θ and D = ejkX sin θ. Note that Equation (6.42) follows directly from

numerical boundary analysis applied to Equation (6.19). However, since an analysis tech-

nique does not have to result in a computable formula in the form of Equation (6.19),


Equation (6.42) can be further rewritten in a simpler equivalent form as

R(z, θ) = −{(

1 +λ

ξw(z)

)z −

[2λ2C + λ2(D + D−1)

+2(1 − 2λ2)]

+(1 − λ

ξw(z)

)z−1

}

/

{(1 +

λ

ξw(z)

)z −

[2λ2C−1 + λ2(D + D−1)

+2(1 − 2λ2)]

+(1 − λ

ξw(z)

)z−1

}. (6.43)

Having obtained a formal expression for the numerical reflectance, it can now be shown

that the 2D boundary represents a passive termination. Since the NBA formula for the

2D digital impedance filter boundary model given by Equation (6.43) reduces to Equation

(5.46) for any complex boundary impedance ξw, such a stability proof is analogous to the

stability proof presented in Section 5.5.2. Such analysis again leads to aw ≥ 0, where aw

is a real part of ξw. Thus the boundary is passive provided that the real part of the digital

wall impedance is nonnegative. This is ensured for any impedance filter that has been

calculated from Equation (6.1) using a digital normal-incidence reflection filter for which

|R0(z)| ≤ 1, and hence the whole simulation is always stable.

6.4.2 3D DIF Model

If we consider a wall normal to the rectangular 3D system x-y-z and located at x = 0,

the discretised total sound pressure in the standing wave in the right boundary plane can

be expressed by Equation (5.61). Next, transforming Equation (5.61) to the z-domain

and writing out the formulae for the pressure variables present in the boundary update


equation given by Equation (6.34), the 3D numerical boundary analysis formula results

Rθ,φ(z) = −{(

1 +λa0

b0

)z −

[λ2(2C + D + D−1

+E + E−1) + 2(1 − 3λ2)]

+ (1 − λa0

b0)z−1

+λ2

b0(C−1 − C) Hw(z)

}

/

{(1 +

λa0

b0

)z −

[λ2(2C−1 + D + D−1

+E + E−1) + 2(1 − 3λ2)]

+ (1 − λa0

b0)z−1

+λ2

b0(C − C−1) Hw(z)

}, (6.44)

where C = ejkX cos θ cos φ, D = ejkX sin θ cos φ, and E = ejkX sin φ. Concerning the passivity

of the 3D DIF model, we can again apply the same procedure as for the 2D case (presented

in Section 5.5.2), which after simple manipulations leads to Equation (5.56), where Q now

has more components. However, it is straightforward to show that Equation (5.56) is

again satisfied for all possible wavenumbers and angles of incidence, and hence the 3D

DIF model is always stable.

6.5 Numerical Experiments

For the investigation of the proposed digital impedance filter boundary formulation for

the standard leapfrog scheme and the comparison with 1D models with KW-pipes, a set

of 2D numerical experiments has been done. In these experiments, the same procedure

has been applied and the same numerical setup used, as described in Section 5.6.1. For

the results in 3D, the results have been obtained with the numerical boundary analysis.

6.5.1 1D Boundary Model

The 2D impedance filter boundary model proposed in this chapter is compared with the

1D boundary model which constitutes a typical termination of 2D and 3D acoustic spaces

in the digital waveguide mesh room acoustics. This 1D FDTD/K-DWM boundary model

is given by [90, 46]

pn+1l,m = [1 + R0] p

nl−1,m − R0p

n−1l,m , (6.45)


where R0 denotes the wall reflectance at normal incidence. By replacing R0 with a digital

filter R0(z), this formulation becomes mathematically equivalent to the 1D termination of

a K-DWM model found in [53, 75]. More details on the 1D reflectance filter and KW-pipes

can be found in Sections 3.2.4 and 3.2.4, respectively.

6.5.2 Impedance Filter Design

In this section, a brief description of the techniques used for the design of the impedance

filters is given. In all cases, the filters were designed to approximate both phase and

amplitude of the boundary impedance for a particular wall type.

Fibrous Material Layer Boundary

The first boundary type under investigation is a layer of fibrous material on a rigid wall,

which is briefly reviewed in Section 2.5. The analytic formulae for the characteristic

impedance Z0(ω) and propagation constant Γ(ω) of the material provided in [5] were used

as a basis to express the (continuous-domain) specific wall impedance ξw(ω) as follows

ξw(ω) =Z0(ω)

ρccoth (Γ(ω)d) , (6.46)

where d is the layer thickness and ρ is the air density. Both Γ and Z0 depend on the

flow resistivity σ; hence the boundary is characterised by the two parameters d and σ.

This type of formulation was originally provided in [29] and later improved on in [5].

The normal-incidence reflectance is obtained from Equation (5.5), which represents the

target response in the design of an appropriate digital reflectance filter; the output-error

minimisation technique [103] was applied here for this purpose, resulting in a 13th-order

IIR filter that closely matches the target response up to about 0.3fs. This filter is then

converted to a digital impedance filter of 13th-order using Equation (6.1).

Mechanical Impedance Boundary

For direct comparison with the model presented in Chapter 5, a second boundary type

was investigated, starting from a continuous-domain mechanical impedance Zw(s) =

ρ c ξw(s) = (Ms2 +Rs+K)/s, where R denotes resistance, M is the mass per unit area, K

denotes the spring constant, and s is the Laplace frequency variable. This impedance was

mapped to the discrete domain using (a) the bilinear transform and (b) the impulse in-

variant method (IIM)1, resulting in two different digital impedance filters of second-order

1The IIM was actually applied to the specific wall admittance, and the resulting digital filter was then

inverted to give the digital impedance. This indirect discretisation is necessary since the IIM is not suited

to mapping high-pass analog filters to the discrete domain.


0 0.05 0.1 0.15 0.2 0.250.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 0o

0 0.05 0.1 0.15 0.2 0.250.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 15o

0 0.05 0.1 0.15 0.2 0.250.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 30o

0 0.05 0.1 0.15 0.2 0.250.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 45o

0 0.05 0.1 0.15 0.2 0.250.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 60o

0 0.05 0.1 0.15 0.2 0.250.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 75o

Figure 6.4: Numerical reflectance of the fibrous material boundary for d = 0.04, σ = 100, and anglesof incidence θ = 0o, 15o, 30o, 45o, 60o, 75o. In each plot, the dashed line shows the 2D DIF boundarymodel, the black solid line shows the 1D filter model, and the solid grey line shows the theoreticalreflectance.

to be investigated.

Low-pass Reflectance Boundary

Finally, we also designed a first-order digital low-pass reflectance filter by applying the IIM

to the reflectance transfer function R(s) = (gα)/(s + α), where g is now a gain factor and

α denotes filter design parameter. The corresponding impedance filter is then obtained

with Equation (6.1).

6.5.3 Results of the 2D DIF Model

In this section, the magnitudes of the experimentally obtained reflectance are analysed

in the frequency domain. In general, the presented figures show the reflectance of the

2D digital impedance filter (DIF) model proposed in this chapter (black dashed lines),

the 1D boundary model that connects reflectance filters to FDTD mesh with the use

of KW-converters (black solid lines), and the theoretical reflection (grey solid lines) up

to a quarter of the sample rate. In order to focus on the FDTD modelling, the initial

filter design approximation error was left out of the comparisons. That is, the theoretical

reflection was taken as the digital reflection for a given digital impedance filter, i.e.

R(z) =ξw(z) cos θ − 1

ξw(z) cos θ + 1. (6.47)


0 0.05 0.1 0.15 0.2 0.250.4

0.5

0.6

0.7

0.8

0.9

1

1.1


Ref

lect

ance

am

plitu

de

Figure 6.5: Numerical reflectance of the fibrous material boundary with d = 0.04 and σ = 100, forθ = 75o. The dashed line shows the reflectance of the 2D DIF boundary model calculated using theNBA, and the solid grey line shows the theoretical reflectance R(z).

Figure 6.4 illustrates the reflectance of the fibrous material layer, where the thickness

of fibres was d = 0.04m and the flow resistivity was assigned an example value σ =

100Nsm−4. The DIF boundary model ‘follows’ the theoretical value quite well for all angles

of incidence. In particular, the numerical reflectance adheres perfectly to the theoretical

reflection for low frequencies at all angles of incidence. Furthermore, a perfect match

is visible for θ = 45o, for which the leapfrog scheme has no dispersion. The numerical

error is the largest at normal incidence, which coincides with the fact that the numerical

dispersion error for the leapfrog scheme is the strongest in axial directions.

In comparison, the 1D boundary model does not adhere well to the theory for most

angles of incidence. Although the numerical reflectance is on average at the correct level,

there is a substantial misalignment along the frequency axis; that is, the minima and

maxima are positioned wrongly for all angles of incidence.

Note that the jumps near (ω = 0) in the reflectance obtained with numerical exper-

iments in all figures, particularly visible at high angles of incidence (e.g. θ = 75o), are

artefacts due only to the wavefront not being perfectly flat. As illustrated in Figure 6.5,

the numerical boundary analysis confirms that a sudden jump near DC for the angle of

incidence θ = 75o does not in fact occur for the presented boundary models.

Regarding the mechanical impedance boundary, the parameters were set to R/(ρc) = 2,

M/(ρcT ) = 6, and KT/(ρc) = 2, where T is the time step. When the bilinear transform

method was used to obtain the digital impedance filter, the simulations with the 1D

boundary model were structurally unstable. No stability problems occurred though when

using the DIF model, and indeed a good match to theory was observed for all incidences.

When using the impulse invariant method, neither boundary model resulted in stability


0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 0o

0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 15o

0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 30o

0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 45o

0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 60o

0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 75o

Figure 6.6: Numerical reflectance of the mechanical boundary discretised using the impulse invariantmethod, for θ = 0o, 15o, 30o, 45o, 60o, 75o. In each plot, the dashed line shows the 2D DIF boundarymodel, the black solid line shows the 1D filter model, and the solid grey line shows the theoreticalreflectance R(z).

problems. However, the 1D boundary model now shows an enormous discrepancy with

theory, while the DIF model adheres extremely well to the theory (see Figure 6.6).

The low-pass reflectance results are depicted in Figure 6.7. As can be seen, the 1D

boundary model performs somewhat better in this case, but is still structurally different

from the theoretical response, and not nearly as accurate as the proposed DIF boundary

model. In addition, since the numerical reflectance of the 1D boundary model is not exact

at ω = 0, there is no scope for improvement in accuracy by up-sampling.

These results show that there are fundamental problems with the 1D boundary model,

and that these are entirely overcome by the proposed DIF model. Furthermore it is

interesting to note that for the DIF model, the perfect match always results for θ = 45o

up to 0.25fs and a weaker match is observed for the normal incidence.

In addition, the 2D NBA formula for the last two boundary filters at normal angle

of incidence is illustrated in Figure 6.8. The reflectance was obtained through numeri-

cal experiments with the setup described in Section 5.6.1, with the difference that the

room interior was modelled with the standard 2D rectilinear scheme, the size of the sim-

ulated space was increased to 2800x2100 and 2800x4200 respectively, the simulation time

increased to 3700 samples, and the distance from the source to the centre of the inves-

tigated wall was set to 100 grid nodes. The perfect match of the numerically simulated

and analytically predicted numerical reflectance proves that NBA accurately predicts the

numerical reflectance of the DIF model.

Finally, the phase of the numerical reflectance is analysed in the frequency domain.


0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 0o

0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1

Normalised frequencyR

efle

ctan

ce a

mpl

itude

θ = 15o

0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 30o

0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 45o

0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 60o

0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 75o

Figure 6.7: Numerical reflectance of the low-pass reflectance boundary with g = 0.85, α = 0.4, andangles of incidence θ = 0o, 15o, 30o, 45o, 60o, 75o. The dashed line shows the reflectance of the 2D DIFboundary model, the black solid line shows the reflectance from the 1D filter model, and the solid greyline shows the theoretical reflectance R(z).

0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

Figure 6.8: Numerical reflectance of the mechanical boundary and the low-pass reflectance boundaryfor θ = 0o. The dashed lines show the reflectance of the 2D DIF boundary model with standard 2Drectilinear scheme room interior implementation, and the solid grey lines show the reflectance predictedwith the NBA method.


0 0.05 0.1 0.15 0.2 0.251

2

3

4

5

6x 10

−3


Pha

se d

elay

[sam

ples

]

θ = 0o

0 0.05 0.1 0.15 0.2 0.251

2

3

4

5

6x 10

−3

Normalised frequencyP

hase

del

ay [s

ampl

es]

θ = 15o

0 0.05 0.1 0.15 0.2 0.251

2

3

4

5

6x 10

−3


Pha

se d

elay

[sam

ples

]

θ = 30o

0 0.05 0.1 0.15 0.2 0.251

2

3

4

5

6x 10

−3


Pha

se d

elay

[sam

ples

]

θ = 45o

0 0.05 0.1 0.15 0.2 0.25

2

4

6

8

x 10−3


Pha

se d

elay

[sam

ples

]

θ = 60o

0 0.05 0.1 0.15 0.2 0.25

2

4

6

8

x 10−3


Pha

se d

elay

[sam

ples

]

θ = 75o

Figure 6.9: The phase delay of the numerical reflectance of the fibrous material boundary for d = 0.04,σ = 100, and angles of incidence θ = 0o, 15o, 30o, 45o, 60o, 75o. In each plot, the dashed line shows the2D DIF boundary model, the black solid line shows the 1D filter model, and the solid grey line showsthe theoretical reflectance.

0 0.05 0.1 0.15 0.2 0.25−1

0

1

2

3x 10

−3


Pha

se d

elay

[sam

ples

]

θ = 0o

0 0.05 0.1 0.15 0.2 0.25−1

0

1

2

3x 10

−3


Pha

se d

elay

[sam

ples

]

θ = 15o

0 0.05 0.1 0.15 0.2 0.25−1

0

1

2

3x 10

−3


Pha

se d

elay

[sam

ples

]

θ = 30o

0 0.05 0.1 0.15 0.2 0.25−1

0

1

2

3x 10

−3


Pha

se d

elay

[sam

ples

]

θ = 45o

0 0.05 0.1 0.15 0.2 0.25−1

0

1

2

3x 10

−3


Pha

se d

elay

[sam

ples

]

θ = 60o

0 0.05 0.1 0.15 0.2 0.25−1

0

1

2

3x 10

−3


Pha

se d

elay

[sam

ples

]

θ = 75o

Figure 6.10: The phase delay of the numerical reflectance of the low-pass reflectance boundary withg = 0.85, α = 0.4, and angles of incidence θ = 0o, 15o, 30o, 45o, 60o, 75o. The dashed line shows thereflectance of the 2D DIF boundary model, the black solid line shows the reflectance from the 1D filtermodel, and the solid grey line shows the theoretical reflectance R(z).


Figures 6.9 and 6.10 illustrate the reflectance phase delay of the fibrous material layer

(d = 0.04m and σ = 100Nsm−4) and the low-pass reflectance (g = 0.85 and α = 0.4),

respectively. In general, the reflectance phase of the 2D digital impedance filter model

adheres well to the phase of theoretical reflectance for all angles of incidence for both

digital impedance filters. In particular, the phase delay is always perfectly matched for at

low frequencies. On the other hand, the phase delay of the 1D boundary model does not

correctly approximate the theoretical phase delay, which confirms that, similarly to the

frequency-independent case presented in Section 5.6.2, a phase shift problem is introduced.

However, note that the phase shift for presented boundaries is in general small. All in all,

it can be concluded that the complex-value numerical error appears in both phase and

amplitude, and depending on the boundary type, can be displayed in either plot in both

ways.

6.5.4 Results of the 3D DIF Model

In this section, results obtained with the use of the 3D numerical boundary analysis are

presented. Similarly to the previous section, all figures illustrate the magnitude of the

numerical reflectance of a 3D DIF boundary model (black dashed lines) compared with the

theoretical reflection (grey solid lines) for up to the cut-off frequency for axial directions,

which for the 3D leapfrog scheme amounts to 0.196fs. The theoretical reflection used as

a reference was again taken as the digital reflection for a given digital impedance filter,

which for a 3D space yields

Rθ,φ(z) =ξw(z) cos θ cos φ − 1

ξw(z) cos θ cos φ + 1. (6.48)

Furthermore, the same three digital impedance filters described in Section 6.5.2 were

tested.

The reflectance of the fibrous material layer for a constant elevation angle φ = 60o and

varying azimuth angles is illustrated in Figure 6.11. The numerical reflectance of the DIF

model exhibits a very good agreement with theory, in particular at low frequencies. Note

that the elevation angle selected for Figure 6.11 was chosen as an example angle; however

a similarly good match was observed for other incidences.

Figure 6.12 illustrates low-pass reflectance of the DIF model for elevation angle φ = 35o.

A perfect match between the numerical and theoretical reflectance occurs for θ = 45o which

coincides with the absence of numerical error of the standard leapfrog scheme in diagonal

directions. Note that in a 3D case the diagonal direction can be defined by a pair of angles

φ ≈ 35.3o and θ = 45o. It can also be observed that numerical reflectance structurally

adheres well to the theory for the remaining azimuth angles.


0 0.05 0.1 0.150.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 0o, φ = 60o

0 0.05 0.1 0.150.4

0.6

0.8

1


efle

ctan

ce a

mpl

itude

θ = 15o, φ = 60o

0 0.05 0.1 0.150.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 30o, φ = 60o

0 0.05 0.1 0.150.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 45o, φ = 60o

0 0.05 0.1 0.150.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 60o, φ = 60o

0 0.05 0.1 0.150.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 75o, φ = 60o

Figure 6.11: Numerical reflectance of the fibrous material boundary for d = 0.04, σ = 100, a constantelevation angle φ = 60o, and the following azimuth angles θ = 0o, 15o, 30o, 45o, 60o, 75o. In eachplot, the black dashed line shows 3D DIF boundary model and the solid grey line shows the theoreticalreflectance.

0 0.05 0.1 0.150

0.2

0.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 0o, φ = 35o

0 0.05 0.1 0.150

0.2

0.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 15o, φ = 35o

0 0.05 0.1 0.150

0.2

0.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 30o, φ = 35o

0 0.05 0.1 0.150

0.2

0.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 45o, φ = 35o

0 0.05 0.1 0.150

0.2

0.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 60o, φ = 35o

0 0.05 0.1 0.150

0.2

0.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 75o, φ = 35o

Figure 6.12: Numerical reflectance of the low-pass reflectance boundary with g = 0.85, α = 0.4, aconstant elevation angle φ = 35o, and the following azimuth angles θ = 0o, 15o, 30o, 45o, 60o, 75o. Theblack dashed line shows the reflectance of the 3D DIF boundary model and solid grey line shows thetheoretical reflectance.


Figure 6.13: Grid spacing ambiguity in 1D terminations of a 2D mesh structure. Boundary nodes areindicated with grey-coloured circles, and black arrows point to the updated node.

These results show that the 3D digital impedance filter boundary model yields an

accurate numerical reflectance for any combination of azimuth and elevation angles, even

for near-grazing incidences.

6.6 Discussion

In the 1D boundary model, the boundary node connects only to the nearest node in the

interior [see Figure 5.2(b)]. Hence at the boundary node, the only direction along which

information is available is normal incidence. In other words, there is a priori no angle

information available that can be incorporated to ensure that the numerical boundary

behaves as a locally reacting surface. The only angle at which it can be hoped for that

the numerical reflectance coincides with the theoretical reflectance is normal incidence.

However, as the results of the numerical experiments show, even that is not the case.

Some light can be shown on the inaccuracies of the 1D boundary model by viewing it

from a FDTD perspective. As has been shown in Section 5.2.2, the 1D FDTD/K-DWM

boundary formula given by Equation (6.45) implies that the distance from the boundary

node to the nearest neighbouring node in the medium is smaller than distances between

room interior nodes. As illustrated in Figure 6.13, when updating an interior node that is

located next to the boundary, the distance between this node and the boundary node is

larger than the distance in the opposite direction, i.e. from the boundary node to the same

interior node when updating the boundary node. Thus the distance between the same two

grid nodes varies depending on which node is updated. This constitutes an ambiguity

about the inter-node spacing at the boundary, probably causing the simulation locally

being consistent with neither the 1D nor the 2D wave equation. While it remains difficult

to precisely analyse or predict the effects of this ambiguity, the numerical experiments in

this chapter indicate severe problems (including instability and large errors) can occur as

a result.

On the other hand, the 2D/3D FDTD digital impedance filter (DIF) formulation in-


terconnects the boundary nodes to both the adjacent node in the interior as well as the

neighbouring nodes at a boundary, thus implementing wave propagation along the wall

surface, and obeying the (discretised) 2D/3D wave equation at the boundary. As such, it

behaves as a locally reacting surface. As with dispersion errors in the standard leapfrog

scheme, the numerical error is most pronounced at the higher end of the frequency axis,

and vanishes altogether for diagonal directions. Indeed, as shown in Figure 6.8, the nu-

merical reflectance of the multidimensional DIF model can be precisely predicted using

methods that are in essence the same as those for analysing dispersion for FDTD schemes.

The numerical results structurally match the magnitude of theoretical reflectance well

for up to the lowest cut-off frequency of the leapfrog scheme approximating the 2D or

3D wave equation (i.e. up to 0.25fs for 2D and up to 0.196fs for 3D boundary model).

In particular, the numerical results always adhere perfectly at low frequencies, which is

a fundamentally useful property of numerical modelling as it allows the improvement in

performance by up-sampling.

Furthermore, the 2D/3D digital impedance filter formulation includes a proper treat-

ment of corners and edges. In the 1D boundary model, the corner is effectively not regarded

as a part of the medium and therefore outer corner nodes are effectively excluded from the

simulation [see Figure 5.2(b)]. This simplification leads to spurious results when a sound

wave reflects from corner terminations of acoustic spaces; for example when a plane wave

travelling in x-direction reflects of a corner, some wave energy is directed in y-direction.

No such problems occur when using the 2D/3D DIF model.

6.7 Conclusions

In this chapter, we have proposed a novel method for constructing numerical formulations

of generalised frequency-dependent boundary model of a locally reacting surface in FDTD

and K-DWM room acoustics simulations by incorporating digital impedance filters (DIF)

in a computable manner. Compact formulations of DIF boundary model were presented

and a good match with the theoretical reflectance magnitude was observed for various

impedance filters and all angles of incidence. Furthermore, a full treatment of corners and

boundary edges is provided.

The proposed DIF boundary formulation represents a significant improvement on the

commonly used 1D boundary model that combines FDTD room interior implementation

with reflectance filters at boundaries with the use of KW-converters, which up to now have

been the only means of modelling generally complex boundaries in non-staggered FDTD

and K-DWM simulations of acoustic spaces. Results obtained with numerical experiments

have indicated that the numerical reflectance of such 1D models can wildly differ from

theory, and may even lead to instability problems. It seems therefore justified to say that


the use of 1D filter models with KW-converters at boundaries should be avoided.

In addition, the performance of the proposed DIF boundary model can be predicted

exactly using the numerical boundary analysis (NBA) method. This is extremely useful, as

it allows researchers and developers to investigate the behaviour of the numerical boundary

without having to run many time-consuming simulations.

158

Chapter 7

Compact Explicit Formulation of

the DIF Model

This chapter discusses the application of the finite difference time-domain (FDTD) method

to modelling 2D and 3D rooms, focusing on scheme-consistent boundary formulations for

compact explicit schemes based on a rectilinear grid. Even though different grid topolo-

gies and higher-order spatial schemes are possible, the proposed boundary models are

constrained to compact schemes on a rectilinear topology since it allows a straightfor-

ward fit of the grid to rooms with parallel walls. The approach presented here constitutes

an extension of the digital impedance filter boundary model formulated for the standard

leapfrog scheme in Chapter 6 to a general family of compact explicit schemes discussed

in Chapter 4. A consistency of scheme in the room interior and at the boundary is very

beneficial since analytic techniques can be applied to analyse and predict the performance

of the model and also to ensure the stability of the whole numerical simulation.

The reader is reminded that the ADI implementation requires explicit computation of

the boundary nodes and implicit computation of the room interior nodes, thus creating

an inconsistency in FDTD scheme between the boundary and the room interior. In some

applications - such as numerical experiments for determining the numerical reflectance of

a boundary model for the standard leapfrog scheme in Chapter 5 - this approach has some

advantages, and does allow obtaining useful and accurate results. However in general any

scheme inconsistency should be avoided if possible, particularly because stability cannot

be guaranteed. For this reason we restrict ourselves here to nonstaggered compact explicit

schemes, the family of which is herein referred to as “interpolated schemes”.

In this chapter, the formulation of a digital impedance filter (DIF) boundary model

for a family of compact explicit FDTD schemes is presented, which is based on the same

principles as the boundary formulation for the standard leapfrog scheme in Section 6, but

requires some additional steps such as interpolation between grid nodes. The issue of

Chapter 7. Compact Explicit Formulation of the DIF Model 159

modelling corners is also discussed, and an analytic prediction of the numerical boundary

reflectance is provided. In the comparison of the 2D results, the focus is on three cases

of particular interest, namely the interpolated wideband (IWB) scheme (that is optimally

efficient in 2D), interpolated isotropic (IISO) scheme, and the interpolated digital waveg-

uide mesh (IDWM). The latter two are nearly isotropic at low frequencies. In comparison

of results for 3D schemes, the octahedral scheme (OCTA) and the tetrahedral scheme

(TETRA) are also investigated in order to cover the whole range of 3D compact explicit

schemes.

The chapter is structured as follows. In Section 7.1 the basic formulation of a digital

impedance filter model is discussed. The full 2D compact explicit digital impedance filter

(IDIF) formulation for boundary nodes and corners is presented in Sections 7.1.1 and

7.1.2, respectively. In Sections 7.1.3 and 7.1.4, the derivation of the 3D compact explicit

digital impedance filter (IDIF) formulation for boundary nodes and corners is respectively

presented. A numerical boundary analysis (NBA) is provided in Section 7.2. A comparison

of boundary reflectance results for the 2D and 3D boundaries is presented respectively in

Sections 7.3 and 7.4, followed by discussion in Section 7.5 and some concluding remarks

in Section 7.6.

7.1 Compact Explicit DIF Formulation

In order to derive the multidimensional FDTD formulation, the discretised boundary

condition has to be combined with the discrete multidimensional wave equation. The DIF

boundary formulations for compact explicit schemes can be derived in a similar way to

the method presented for the standard leapfrog scheme in Chapter 6, by combining the

boundary condition with the respective scheme approximating the wave equation. This

procedure is consistent with locally reacting surface theory provided that the boundary

condition is applied across the boundary in the direction normal to the boundary.

7.1.1 2D Compact Explicit DIF Boundary Model

If we consider a right boundary for the 2D compact explicit FDTD scheme given by

Equation (4.8), one notices immediately that there are three points (i.e. nodes) lying out

of the modelled space, as depicted in Figure 7.1. As mentioned in previous chapters, these

are often referred to as ‘ghost points’ or ‘ghost nodes’ [113], and they can be eliminated

from the compact explicit scheme equation with the use of the boundary condition. The

derivation of the compact explicit scheme boundary formulation can be divided in two

elimination steps: 1) for the axial ghost point and 2) for the two diagonal ghost points.

The first elimination is exactly the same as in the derivation of the digital impedance

filter boundary for the standard leapfrog scheme presented in Section 6.2, and starts


l,m-1p

l-1,mp

l,mp

l,m+1p

l+1,mp

l+1,m+1p

l+1,m-1p

l-1,m-1p

l-1,m+1p

l-1,mp

l+1,mp~ ~

Figure 7.1: The interpolated mesh at a right boundary. Ghost nodes are indicated with white-colouredcircles, room interior nodes are indicated with black-coloured circles, interpolated nodes are indicatedwith squares, and the room interior is indicated by grey shading.

by approximating the first-order derivatives in time and space domains of the continuous

boundary condition given by Equation (5.4) with centered finite difference operators. Next,

the discrete boundary condition is transformed to the z-domain and the impedance filter

is defined as

ξw(z) =b0 + B(z)

a0 + A(z)(7.1)

where the filter nominator and denominator are given as

B(z) =

N∑

i=1

biz−i, A(z) =

N∑

i=1

aiz−i. (7.2)

Further manipulations include rewriting this condition in terms of an intermediate value

so that a computable boundary condition results, which after applying an inverse z-

transformation yields an explicit filter difference equation. The final update formula for

an axial ghost point pl+1,m of a right boundary is given as

pnl+1,m = pn

l−1,m +a0

λb0(pn−1

l,m − pn+1l,m ) +

gna

b0, (7.3)

with the explicit filter difference equation updated according to

yna =

1

a0

(b0 xn

a + gna

), (7.4)

where an intermediate value ga is computed using past filter values only according to

gna =

N∑

i=1

(bi xn−i

a − ai yn−ia

), (7.5)


and filter input xa at time step n is defined as

xna = pn

l−1,m − pnl+1,m =

a0

λb0(pn+1


a

b0. (7.6)

Subscript a in Equations(7.3) - (7.6) denotes the boundary condition in axial direction.

The second elimination concerns the two diagonal ghost points. First we apply linear

interpolation on the diagonal points lying inside and outside of the modelled space, which

can be expressed as

p nl−1,m =

1

2(pn

l−1,m+1 + pnl−1,m−1), (7.7)

p nl+1,m =

1

2(pn

l+1,m+1 + pnl+1,m−1), (7.8)

where pl−1,m and pl+1,m are the pressure values at two interpolated nodes lying on the

circle that goes through the four diagonal mesh points. Hence these interpolated nodes are

equally distant1 from the boundary node pl,m, as depicted in Figure 7.1. The interpolated

values are located across the boundary in the direction normal to the wall and thus can

be used in the application of the discrete boundary condition

p nl+1,m = p n

l−1,m +a0

λb0(pn−1

l,m − pn+1l,m ) +

gnd

b0. (7.9)

Similarly to the axial direction, the explicit filter formula for the diagonal direction (de-

noted with subscript d) then reads

ynd =

1

a0

(b0 xn

d + gnd

), (7.10)

where an intermediate value gd yields

gnd =

N∑

i=1

(bi xn−i

d − ai yn−id

), (7.11)

and filter input xd at time step n is defined as

xnd = p n

l−1,m − p nl+1,m =

a0

λb0(pn+1


d

b0. (7.12)

Next, Equation (7.9) can be expressed in terms of the points lying on the original rectilinear

1Interpolation for points positioned less far from the central point than X leads to stability problems,

because the stability condition is effectively violated in that case.


grid as

1

2(pn

l+1,m+1 + pnl+1,m−1)=

1

2(pn

l−1,m+1 + pnl−1,m−1)

+a0

λb0(pn−1

l,m − pn+1l,m ) +

gnd

b0. (7.13)

If we now substitute for the three ghost points in the interpolated FD scheme approximat-

ing the 2D wave equation with the two boundary subconditions given by Equations (7.3)

and (7.13), and apply some simple algebraic manipulations, the following formula results

pn+1l,m =

[λ2(1 − 2b)(2pn


l,m−1 +gna

b0)

+λ22b(pnl−1,m+1 + pn

l−1,m−1 +gnd

b0) + 2(1 − 2λ2

+2λ2b)pnl,m + (

λa0

b0− 1)pn−1

l,m

]/(1 +

λa0

b0

). (7.14)

The update formula given by Equation (7.14) would require the implementation of two

boundary filters for one boundary node. However, both subconditions use the same

impedance filter with a difference in filter input and output signals; the latter is scaled

by a different factor when returned in Equation (7.14). Therefore, we can next rewrite

Equation (7.14) as

pn+1l,m =

[λ2(1 − 2b)(2pn


l,m−1)

+λ22b(pnl−1,m+1 + pn

l−1,m−1) +λ2

b0gn + 2(1 − 2λ2

+2λ2b)pnl,m + (

λa0

b0− 1)pn−1

l,m

]/(1 +

λa0

b0

), (7.15)

where the terms including intermediate filter values for axial and diagonal directions are

grouped into one intermediate filter value g, which yields

gn = (1 − 2b) gna + 2b gn

d . (7.16)

From the definition of a linear system, which a digital impedance filter obviously is, the

relationship between system input and output satisfies the superposition property. Hence

we can next move the weighting coefficients originally located in front of intermediate filter

variables gna and gn

d , to the front of the filter input values as

xn = (1 − 2b) xna + 2b xn

d (7.17)

in order to obtain the final boundary update formula in terms of one boundary filter only.


After some simple algebraic manipulations, the final filter input x equation yields

xn =a0

λb0(pn+1


b0, (7.18)

and the explicit filter difference equation is given as

yn =1

a0

(b0 xn + gn

), (7.19)

where intermediate value g is updated according to

gn =

N∑

i=1

(bi xn−i − ai yn−i

). (7.20)

Thus three filter equations (7.18), (7.19) and (7.20) are the same as filter equations used

for the standard leapfrog scheme in Chapter 6.

The DIF boundary formulation for the general compact explicit 2D FDTD scheme

requires updating the intermediate value g, the right boundary node pl,m, filter input and

output signals at each time step according to Equations (7.20), (7.15), (7.18) and (7.19),

respectively. Note that computing these equations in the given order avoids the necessity

to update ghost nodes and that one IIR filter is needed for each boundary node. Such a

formulation can be used as long as b0 6= 0 and b0 6= 1.

7.1.2 2D Compact Explicit DIF Corners

Corner nodes are part of the medium, and therefore update formulae must be applied at

these nodes as well. For the derivation of an update formula for an outer corner, the 2D

compact explicit FDTD scheme equation (4.8) is combined with two boundary conditions

that have to be satisfied simultaneously. For a right-top boundary corner, the continuous

boundary conditions have in x- and y-direction are respectively given by Equation (5.13).

The additional condition∂p2

∂x∂y= 0, (7.21)

holds at the corner, which guarantees local coherence between the two meeting boundaries

[72]. In the discrete form this conditions reads

pnl+1,m+1 = pn

l+1,m−1 + pnl−1,m+1 − pn

l−1,m−1. (7.22)

As depicted in Figure 7.2(a), there are originally five ghost points to eliminate in the dis-

crete wave equation. Similarly to the boundary model derivation, the discretised boundary

condition in x-direction can be split into two subconditions: 1) for an axial node given by


l,m-1p

l-1,mp

l,mp

l,m+1p

l+1,mp

l+1,m+1p

l+1,m-1p

l-1,m-1p

l-1,m+1p

l-1,mp l+1,m

p~ ~

l,m+1p~

l,m-1p~

l,m-1p

l-1,mp

l,mp

l,m+1p

l+1,mp

l+1,m+1p

l+1,m-1p

l-1,m-1p

l-1,m+1p

(a) (b)

Figure 7.2: The interpolated mesh at: (a) top-right outer corner and (b) inner corner. Ghost nodesare indicated with white-coloured circles, room interior nodes are indicated with black-coloured circles,interpolated nodes are indicated with squares, and the room interior is indicated by grey shading.

Equation (7.3) and 2) for two diagonal nodes given by Equation (7.9). The latter one is

next combined with Equation (7.22) in order to eliminate for the ghost point pl+1,m+1 in

a boundary subcondition for diagonal nodes in x-direction, which yields

pnl+1,m−1 = pn

l−1,m−1 +ax

λbx(pn−1

l,m − pn+1l,m ) +

gndx

bx, (7.23)

where the intermediate value for the diagonal x-direction gdx is computed directly from

impedance filter implementation, bx and ax are the b0 and a0 values for the boundary

condition in x-direction. The same procedure applies to the boundary condition for the

interpolated diagonal pressure values in y-direction.

After some mathematical manipulations, which include initially applying Equation

(7.22) to the discrete 2D wave equation approximated with the compact explicit scheme,

and next substituting for the ghost points with Equations(7.3) and (7.23) in x-direction

and their respective counterparts in y-direction, the final update formula for the right-top

outer corner node is

pn+1l,m =

[λ2(1 − 2b)(2pn

l−1,m + 2pnl,m−1) + λ24b pn

l−1,m−1

+λ2(gnx

bx+

gny

by) + 2(1 − 2λ2 + 2λ2b)pn

l,m

+(λax

bx+

λay

by− 1)pn−1

l,m

]/(1 +

λax

bx+

λay

by

), (7.24)

where gx and gy are computed from their respective filter implementations, bx and by

are the filter b0 values, and ax and ay are the filter a0 values for the x- and y-direction


boundary condition, respectively. Note that such a corner formulation requires computing

the outer corner node pl,m with an update formula given by Equation (7.24) and updating

two filter input signals in x- and y-directions according to the following update formulae

xnx =

ax

λbx(pn+1


x

bx, (7.25)

xny =

ay

λby(pn+1

l,m − pn−1l,m ) −

gny

by. (7.26)

At inner corners, neither of the two boundary conditions apply. This implies that the

formula for interior nodes [given by Equation (4.8)] can be used to update the node there.

However, the pressure node pl+1,m+1 is in this case located outside the modelled space,

as depicted in Figure 7.2(b). We can eliminate this ghost point by again applying the

coherence condition given by Equation (7.22), which yields the update formula

pn+1l,m = λ2(1 − 2b)(pn

l+1,m + pnl−1,m + pn

l,m+1 + pnl,m−1)

+λ22b(pnl+1,m−1 + pn

l−1,m+1)

+2(1 − 2λ2 + 2λ2b)pnl,m − pn−1

l,m . (7.27)

Throughout the thesis, it is assumed that boundaries can be mapped on a rectilinear

grid in such a way that only walls parallel to the axes of the coordinate system result,

thus omitting all cases in which boundaries meet at obtuse or acute angles. The cases

omitted would add a large number of possible boundary and corner formulations, adding

significantly to the implementation complexity. It seems much simpler to approximate

angled and curved boundaries by a staircase approximation. In addition, the formulation

of the omitted cases could lead to inconsistencies in the numerical reflectance, especially

so at corner points.

7.1.3 3D Compact Explicit DIF Boundary Model

The derivation of the right boundary condition for the general 3D compact explicit FDTD

scheme (herein also referred to as the interpolated scheme) given by Equation (4.49)

involves eliminating nine ghost points, as illustrated in Figure 7.3. This time we can

differentiate three groups of nodes lying out of the modelled space, and hence the general

boundary condition for the 3D interpolated schemes is divided into three subconditions.

These elimination steps include substituting for: 1) the axial ghost point, 2) the four

side-diagonal ghost points and 3) the four diagonal ghost points, respectively.

The axial ghost point is eliminated in the same way as in the 3D DIF boundary model


for the standard leapfrog scheme presented in Section 6.3, which reads

pnl+1,m,i = pn

l−1,m,i +a0

λb0(pn−1


gna

b0, (7.28)

where l, m, and i denote spatial indexes in x-, y-, and z-direction, respectively. The filter

difference equation ya, an intermediate value ga, and filter input xa are given respectively

by Equations (7.4), (7.5), and (7.6), and subscript a denotes the boundary condition in

axial direction.

The second elimination concerns the four side-diagonal ghost points. First we apply

linear interpolation on the side-diagonal points lying inside and outside of the modelled

space, which can be expressed as

p nl−1,m,i =

1

4(pn

l−1,m+1,i + pnl−1,m−1,i + pn

l−1,m,i+1 + pnl−1,m,i−1), (7.29)

p nl+1,m,i =

1

4(pn

l+1,m+1,i + pnl+1,m−1,i + pn

l+1,m,i+1 + pnl+1,m,i−1), (7.30)

where pl−1,m,i and pl+1,m,i are the pressure values at two interpolated nodes lying on the

sphere that goes through all eight side-diagonal ghost points, as depicted in Figure 7.3(b).

Similarly to the 2D case, these interpolated nodes are equally distant from the boundary

node pl,m,i so that the stability condition for a family of compact explicit schemes is

still obeyed. Since the interpolated values are located across the boundary, the discrete

boundary condition in the direction normal to the wall can be applied, which yields

p nl+1,m,i = p n

l−1,m,i +a0

λb0(pn−1


gnsd

b0, (7.31)

where the explicit filter equations for the side-diagonal direction (denoted with subscript

sd) are respectively given as

ynsd =

1

a0

(b0 xn

sd + gnsd

), (7.32)

gnsd =

N∑

i=1

(bi xn−i

sd − ai yn−isd

), (7.33)

and

xnsd = p n

l−1,m − p nl+1,m =

a0

λb0(pn+1


sd

b0. (7.34)

Equation (7.28) written explicitly in terms of the points lying on the original rectilinear


grid yields

1

4(pn

l+1,m+1,i + pnl+1,m−1,i

+pnl+1,m,i+1 + pn

l+1,m,i−1)=1

4(pn

l−1,m+1,i + pnl−1,m−1,i

+pnl−1,m,i+1 + pn

l−1,m,i−1)

+a0

λb0(pn−1


gnsd

b0. (7.35)

The third elimination step is similar to the previous one, and consists in first applying

linear interpolation on the four diagonal ghost points lying inside and outside of the

modelled space, which is given as

p nl−1,m,i =

1

4(pn

l−1,m+1,i+1 + pnl−1,m−1,i+1 + pn

l−1,m+1,i−1 + pnl−1,m−1,i−1), (7.36)

p nl+1,m,i =

1

4(pn

l+1,m+1,i+1 + pnl+1,m−1,i+1 + pn

l+1,m+1,i−1 + pnl+1,m−1,i−1), (7.37)

where pl−1,m,i and pl+1,m,i are the pressure values at two interpolated nodes lying on the

sphere that goes through all eight diagonal ghost points, as illustrated in Figure 7.3(c).

The resulting boundary condition in terms of the two interpolated values yields

p nl+1,m,i = p n

l−1,m,i +a0

λb0(pn−1


gnd

b0, (7.38)

and in terms of the nodes lying on the original rectilinear grid, it reads

1

4(pn

l+1,m+1,i+1 + pnl+1,m−1,i+1

+pnl+1,m+1,i−1 + pn

l+1,m−1,i−1)=1

4(pn

l−1,m+1,i+1 + pnl−1,m−1,i+1

+pnl−1,m+1,i−1 + pn

l−1,m−1,i−1)

+a0

λb0(pn−1


gnd

b0, (7.39)

where subscript d denotes the diagonal direction. The filter input is defined as

xnd = p n

l−1,m,i − p nl+1,m,i =

a0

λb0(pn+1

l,m,i − pn−1l,m,i) −

gnd

b0, (7.40)

and the filter output yd and intermediate value gd are given respectively by Equations

(7.10) and (7.11).

Finally, the substitution for the nine ghost points in the interpolated FD scheme ap-

proximating the 3D wave equation with the three boundary subconditions given respec-

tively by Equations (7.28), (7.35), and (7.39) yields the update formula for the boundary


node

pn+1l,m,i =

[λ2(1 − 4b + 4c)(2pn


l,m−1,i

+pnl,m,i+1 + pn

l,m,i−1)

+λ22(b − 2c)(pn

l−1,m+1,i + pnl−1,m−1,i + pn

l−1,m,i+1 + pnl−1,m,i−1

)

+λ2(b − 2c)(pn

l,m+1,i+1 + pnl,m+1,i−1 + pn

l,m−1,i+1 + pnl,m−1,i−1

)

+λ22c(pn

l−1,m+1,i+1 + pnl−1,m−1,i+1 + pn

l−1,m+1,i−1 + pnl−1,m−1,i−1

)

+λ2

b0gn + 2(1 − 3λ2 + λ26b − λ24c)pn

l,m,i

+(λa0

b0− 1)pn−1

l,m,i

]/(1 +

λa0

b0

). (7.41)

Since all three subconditions use the same impedance filter with a difference in filter input

and output signals, the overall intermediate filter value g is obtained by the following

grouping

gn = (1 − 4b + 4c) gna + (4b − 8c) gn

sd + 4c gnd . (7.42)

Using the superposition property of a linear system, the weighting coefficients located in

front of intermediate filter variables gna , gn

sd, and gnd , are next moved to the front of the

filter input values as

xn = (1 − 4b + 4c) xna + (4b − 8c) xn

sd + 4c xnd (7.43)

so that only one boundary filter is used per boundary node. Consequently, it is not

necessary to compute the intermediate and filter input values from Equations (7.42) and

(7.43). Instead, the DIF boundary formulation for the 3D interpolated FDTD scheme

requires updating the intermediate value g, the right boundary node pl,m,i, filter input

and output signals at each time step according to Equations (7.20), (7.41), (7.18) and

(7.19), respectively. Thus the filter implementation is analogous to the 2D case and it

is independent of the choice of the scheme, only the update formula for the boundary

node depends on the scheme. Again, computing the above values in that order avoids the

necessity to update the ghost points.

7.1.4 3D Compact Explicit DIF Corners

For derivation of an update formula for a front-top-right outer corner node of a 3D system,

the discrete 3D wave equation is combined with three boundary conditions, which in the


(l+1,m+1,i-1)

( l+1,m,i-1)

(l-1,m-1,i-1)

(l-1,m-1,i)

(l-1,m-1,i+1)

( l+1,m-1,i-1)(l,m-1,i-1)

(l+1,m+1,i-1)

( l+1,m,i-1)

(l-1,m-1,i-1)

(l-1,m-1,i)

( l-1,m-1,i+1)

( l+1,m-1,i-1)(l,m-1,i-1)

(a) (b)

(l+1,m+1,i-1)

( l+1,m,i-1)

(l-1,m-1,i-1)

(l-1,m-1,i)

( l-1,m-1,i+1)

( l+1,m-1,i-1)(l,m-1,i-1)

(c)

Figure 7.3: The 3D compact explicit mesh at a right boundary: (a) with original grid points, (b) withside-diagonal interpolation, and (c) with diagonal interpolation. Ghost nodes are indicated with white-coloured circles, room interior nodes are indicated with black-coloured circles, interpolated nodes areindicated with squares, and the room interior is indicated by grey shading.

continuous space-time domain are expressed as

∂p

∂t= −c ξx

∂p

∂x,

∂p

∂t= −c ξy

∂p

∂y,

∂p

∂t= −c ξz

∂p

∂z(7.44)

where ξx, ξy, and ξz denote digital impedance filters in x-, y- and z-direction, respectively.

Analogous to the 2D case, the additional condition holds at the corner for local coherence

between the meeting boundaries, namely

∂p3

∂x∂y∂z= 0. (7.45)

Some additional conditions may also be used in the three directions for local coherence at

a corner, which yields

∂p2

∂x∂y=0,

∂p2

∂x∂z=0,

∂p2

∂y∂z=0. (7.46)

The discrete forms of conditions governed by Equations (7.44), (7.45), and (7.46) are next

applied to substitute for the ghost points in the 3D compact explicit scheme equation

(4.49).


As depicted in Figure 7.4(a), there are originally nineteen ghost points to eliminate

in the discrete wave equation. Similarly to the derivation of the 3D boundary model, the

discretised boundary condition in x-direction can be split into three subconditions: 1) for

an axial node given by Equation (7.28), 2) for four side-diagonal nodes given by Equation

(7.35), and 3) for four diagonal nodes given by Equation (7.39). The first subcondition is

directly applied in the discrete wave equation. The second subcondition is combined with

the first two equations in (7.46) and is next applied to the discrete wave equation. The

third subcondition in combination with the discrete version of Equation (7.45) is used to

eliminate for diagonal points in the x-direction. The same procedure also applies to the

boundary conditions in y and z-direction, respectively.

After some mathematical manipulations, the final update formula for the front-right-

top outer corner node of a 3D acoustic space is given as

pn+1l,m,i =

[λ22(1 − 4b + 4c)(pn

l−1,m,i + pnl,m−1,i + pn

l,m,i−1)

+λ24(b − 2c)(pn

l−1,m−1,i + pnl−1,m,i−1 + pn

l,m−1,i−1

)

+λ28cpnl−1,m−1,i−1

+λ2(gn

x

bx+

gny

by+

gnz

bz

)+ 2(1 − 3λ2 + λ26b − λ24c)pn

l,m,i

+(λax

bx+

λay

by+

λaz

bz− 1)pn−1

l,m,i

]/(1 +

λax

bx+

λay

by+

λaz

bz

). (7.47)

where gx, gy, and gz are computed from their respective filter implementations, bx, by, and

bz are the filter b0 values, and ax, ay, and az are the filter a0 values for the x-, y-, and

z-direction boundary condition, respectively. The respective filter input signals are given

in the form of Equations (7.25) and (7.26) for the 2D case.

At inner top-right corner of a 3D space, there is only one pressure node pl+1,m+1,i+1

that is located outside the modelled space, as depicted in Figure 7.4(b). We can eliminate

this ghost point from the 3D compact explicit scheme equation by again applying the

discrete version of the coherence condition given by Equation (7.45), which yields the final

update formula for an inner corner node

pn+1l,m,i = λ2(1 − 4b + 4c)

(pn


l,m+1,i + pnl,m−1,i + pn

l,m,i+1 + pnl,m,i−1

)

+λ2(b − 2c)(pn

l+1,m+1,i + pnl+1,m−1,i + pn

l+1,m,i+1 + pnl+1,m,i−1

+ pnl,m+1,i+1 + pn

l,m+1,i−1 + pnl,m−1,i+1 + pn

l,m−1,i−1

+ pnl−1,m+1,i + pn

l−1,m−1,i + pnl−1,m,i+1 + pn

l−1,m,i−1

)

+λ22c(pn

l+1,m−1,i+1 + pnl+1,m+1,i−1 + pn

l−1,m+1,i+1 + pnl−1,m−1,i−1

)

+d4 pnl,m − pn−1

l,m . (7.48)


(a) (b)

Figure 7.4: The 3D compact explicit mesh at: (a) an outer corner, (b) an inner corner. Ghost nodesare indicated with white-coloured circles, room interior nodes are indicated with black-coloured circles,and the room interior is indicated by grey shading.


Numerical boundary analysis (NBA) is a technique proposed in Section 5.5 for exact

prediction of the numerical reflectance of FDTD multidimensional boundary models. For

2D compact explicit schemes, this procedure - a detailed explanation of which can be

found in Sections 5.5 and 6.4 - yields the numerical reflectance

Rθ(z) = −{(

1 +λ

ξw(z)

)z +

(1 − λ

ξw(z)

)z−1

−[λ2(1 − 2b)(2C + D + D−1)

+λ22b C(D + D−1) + 2(1 − 2λ2 + 2λ2b)]}

/

{(1 +

λ

ξw(z)

)z +

(1 − λ

ξw(z)

)z−1

−[λ2(1 − 2b)(2C−1 + D + D−1)

+λ22b C−1(D + D−1) + 2(1 − 2λ2 + 2λ2b)]}

,

(7.49)

where z denotes the z-transform variable, C = ejkX cos θ, D = ejkX sin θ, and k is the

numerical wavenumber, that has to be computed from the general dispersion relation for

compact explicit schemes given by Equation (4.37).

Similarly, the formula for numerical reflectance for the 3D compact explicit schemes

is derived in an analogous way to the derivations presented in Sections 5.5 and 6.4, which


yields

Rθ,φ(z) = −{(

1 +λ

ξw(z)

)z +

(1 − λ

ξw(z)

)z−1

−[λ2(1 − 4b + 4c)(2C + D + D−1 + E + E−1)

+2λ2(b − 2c) C(D + D−1 + E + E−1)

+λ2(b − 2c)(DE + D−1E + DE−1 + D−1E−1)

+2cλ2 C(DE + D−1E + DE−1 + D−1E−1)

+2(1 − 3λ2 + 6bλ2 − 4cλ2)]}

/

{(1 +

λ

ξw(z)

)z +

(1 − λ

ξw(z)

)z−1

−[λ2(1 − 4b + 4c)(2C−1 + D + D−1 + E + E−1)

+2λ2(b − 2c) C−1(D + D−1 + E + E−1)

+λ2(b − 2c)(DE + D−1E + DE−1 + D−1E−1)

+2cλ2 C−1(DE + D−1E + DE−1 + D−1E−1)

+2(1 − 3λ2 + 6bλ2 − 4cλ2)]}

,

(7.50)

where C = ejkX cos θ cos φ, D = ejkX sin θ cos φ, and E = ejkX sinφ. The numerical wavenum-

ber k is computed from Equation (4.64) for the respective compact explicit scheme.

Apart from the analytic reflectance prediction, the NBA also lends itself to proving the

stability of the FDTD boundary formulation, by showing that the boundary represents

a passive termination. As explained in Section 5.5, passivity of an FDTD boundary

formulation is sufficient but not necessary condition for numerical stability and means

that all of the system’s internal modes are stable. Such a stability proof is presented here

for the 2D case; an analogous proof for the 3D compact explicit schemes is omitted for

brevity.

Let us first rewrite Equation (7.49) as

Rθ(z) = − Q − F C

Q − F C−1, (7.51)

where for any numerical wavenumber such that −π/X ≤ k ≤ π/X, the new variable Q


can be expressed as

Q=[2 cos(ωT ) − λ2(1 − 2b) cos(kX sin θ)

−2(1 − 2λ2 + 2λ2b)]+ j

[ √2

ξw(z)sin(ωT )

](7.52)

and the new variable F denotes

F = λ22(1 − 2b) + λ24b cos(kX sin θ). (7.53)

The digital impedance filter transfer function is generally complex-valued, hence for any

one particular frequency we may define the impedance as ξw(z) = aw + j bw. The multi-

plicative reciprocal of such a complex impedance reads

1

ξw=

aw

a2w + b2

w

− jbw

a2w + b2

w

, (7.54)

which now defines the variable Q as

Q=[2 cos(ωT ) − λ2(1 − 2b) cos(kX sin θ)

−2(1 − 2λ2 + 2λ2b) +2λbw

a2w + b2

w

sin(ωT )]

+j[ 2λaw

a2w + b2

w

sin(ωT )]. (7.55)

The boundary model is passive if there are no growing solutions of the system with respect

to time and is given by the following condition∣∣∣Rθ(z)

∣∣∣ ≤ 1, which from Equation (7.51)

can be rewritten as

|Q − F C| ≤∣∣Q − F C−1

∣∣ . (7.56)

Next, taking the square of the left-hand and the right-hand side of Equation (7.56) one

obtains

|Q − F C|2={

Re{Q} − F cos(kX cos θ

)}2

+{

Im{Q} − F sin(kX cos θ

)}2, (7.57)

∣∣Q − F C−1∣∣2={

Re{Q} − F cos(kX cos θ

)}2

+{

Im{Q} + F sin(kX cos θ

)}2, (7.58)

where Re{Q} and Im{Q} are the real and imaginary parts of Q, respectively. Inserting


Equations (7.57) and (7.58) into Equation (7.56) and applying some simple algebraic

manipulations, the following condition results

− Im{Q} F sin(kX cos θ

)≤ Im{Q} F sin

(kX cos θ

). (7.59)

Because the following conditions are met:

1) sin(ωT ) ≥ 0 for frequencies up to Nyquist,

2) sin(kX cos θ

)≥ 0 for −π/2 < θ ≤ π/2,

3) cos(kX sin θ

)≥ 0 for −π/2 < θ ≤ π/2,

4) λ22(1 − 2b) ≥ 0 and λ24b ≥ 0 since λ ≥ 0 and 0 ≤ b ≤ 12 ,

Equation (7.59) can be finally reduced to

aw ≥ 0. (7.60)

Therefore, the boundary termination is passive provided that the real part of the digital

wall impedance is nonnegative, which is guaranteed for any physically feasible impedance

filter that has been calculated using a digital normal-incidence reflection filter for which

|R0(z)| ≤ 1. A similar stability proof can be carried out for corners, but is left out here

for brevity. Due to the fact that the medium itself is lossless (i.e. interpolated FDTD

schemes approximating the 2D wave equation that are presented in this thesis do not

cause numerical attenuation), it follows that the simulation as a whole is always stable.

7.3 2D Boundary Performance

In this section, the reflectance magnitude of the 2D compact explicit DIF boundary model

is analysed in the frequency domain for three schemes, namely the interpolated digi-

tal waveguide mesh, interpolated isotropic, and interpolated wideband scheme. All fig-

ures compare the numerical reflectance obtained from numerical boundary analysis (black

dashed lines) to the theoretical reflectance (grey solid lines). Since the three schemes all

have a different numerical cutoff, the results are provided for the full bandwidth (i.e. up

to the Nyquist frequency 0.5fs). The comparisons are restricted to two angles of incidence,

namely θ = 0o and θ = 45o, as they are the most representative in terms of the numerical

reflectance error and cutoff frequency (i.e., they represent best and worst approximation

cases).

7.3.1 Frequency-independent Results

Frequency-independent results are particularly informative as they give an immediate pic-

ture of the performance for a range of impedances up to the respective cutoff frequencies.


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1


efle

ctan

ce a

mpl

itude

θ = 0o

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

1.2


Ref

lect

ance

am

plitu

deθ = 45o(a)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 0o

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

1.2


Ref

lect

ance

am

plitu

de

θ = 45o(b)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 0o

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

1.2


Ref

lect

ance

am

plitu

de

θ = 45o(c)

Figure 7.5: Numerical reflectance amplitude (dashed black lines) compared to theoretical reflectanceamplitude (solid grey lines) for the following angles of incidence θ = 0o and 45o, and real specific acousticimpedances ξw = 3

2, 7

3, 4, 9, 10000 for: (a) the 2D interpolated digital waveguide mesh boundary, (b)

the 2D interpolated isotropic boundary, (c) the 2D interpolated wideband boundary.


Figure 7.5 illustrates the magnitude of the numerical reflectance of the three interpo-

lated schemes, respectively, for the following real values of specific acoustic impedance

ξw = 32 , 7

3 , 4, 9, and 10000. A general observation can be made that the lower the impedance

value (and hence the resulting reflectance) the more pronounced the numerical error be-

comes. As illustrated in Figure 7.5(a), the numerical reflectance error of the IDWM DIF

boundary is strongly pronounced for both angles of incidence and the cutoff frequencies

are relatively close to each other. The numerical error is small for up to 0.11fs and it is

almost isotropic for frequencies up to around 0.22fs but with a high overall error. Figure

7.5(b) shows that the IISO boundary has higher cutoff frequencies than the IDWM, and

thus the reflectance is generally more accurate for a wider frequency range. More pre-

cisely, the IISO DIF model is nearly isotropic with a relatively small overall error for up

to around 0.27fs and there is almost no error for frequencies up to 0.19fs. As depicted in

Figure 7.5(c), the IWB DIF model has a constant cutoff in all directions that falls at the

Nyquist frequency, which implies that such a boundary provides a full bandwidth of the

numerical simulation at all incidences. The numerical reflectance matches theory perfectly

at normal incidence for all frequencies, and the numerical error in axial direction (where

it is very similar to the IISO boundary) can be considered insignificantly small for up to

almost 0.26fs.

7.3.2 Frequency-dependent Results

Concerning the performance of the frequency-dependent compact explicit DIF boundary,

an example reflectance filter is designed to approximate both phase and amplitude of a

theoretical normal-incidence reflectance. The investigated boundary is a layer of fibrous

material on a rigid wall, for which the continuous-domain specific wall impedance is calcu-

lated from the analytic formulae provided in Section 2.5. For a given analytic impedance,

the digital impedance filter of 15th-order is designed to match closely the target response

for frequencies up to 0.5fs (for a detailed procedure see Section 6.5.2). In addition, the

theoretical reflectance is defined here as the digital reflectance for a given impedance filter

so that the initial filter design error is eliminated from the comparisons.

Figure 7.6 illustrates the magnitude of the numerical reflectance of the fibrous material

layer for the three interpolated schemes, where the thickness of fibres is d = 0.06m and the

flow resistivity equals σ = 200Nsm−4. A general observation can be made that the maxima

and minima of the numerical reflectance are always positioned correctly for θ = 45o,

and that there is a general trend of exaggeration of the variation in amplitude at higher

frequencies. At normal incidence, the IDWM and the IISO boundary are accurate at low

frequencies, but the frequency axis is compressed, resulting in the same cutoff frequencies

as for the respective schemes for updating the interior nodes. The IWB DIF boundary


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1


efle

ctan

ce a

mpl

itude

θ = 0o

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

1.2


Ref

lect

ance

am

plitu

deθ = 45o(a)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 0o

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

1.2


Ref

lect

ance

am

plitu

de

θ = 45o(b)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1


Ref

lect

ance

am

plitu

de

θ = 0o

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

1.2


Ref

lect

ance

am

plitu

de

θ = 45o(c)

Figure 7.6: Numerical reflectance amplitude of the fibrous material boundary with d = 0.04 and σ = 200(dashed black lines) compared to theoretical reflectance amplitude (solid grey lines) for the followingangles of incidence θ = 0o and 45o, for: (a) the 2D interpolated digital waveguide mesh boundary, (b)the 2D interpolated isotropic boundary, (c) the 2D interpolated wideband boundary.


model yields a perfect approximation in both phase and amplitude for θ = 0o for the full

bandwidth, and is also the most accurate of all three schemes for θ = 45o. The analysis of

accuracy and isotropy yields similar frequency ranges to the frequency-independent case.

7.4 3D Boundary Performance

This section presents the reflectance amplitude results for a few special cases of the 3D

compact explicit schemes obtained with the use of the numerical boundary analysis (black

dashed lines) and plotted against the theoretical reflectance (grey solid lines). The pre-

sented boundaries include the digital impedance filter models for the following 3D schemes:

the interpolated digital waveguide mesh, the octahedral, the tetrahedral, the interpolated

wideband, and the interpolated isotropic schemes. All the plots are presented up to the

Nyquist frequency so that the differences in cutoff frequencies for various schemes can

be compared. Since the best and worst performance always results for one of the three

extreme directions, the boundary performance is restricted to the following angles of in-

cidence: the normal incidence (θ = 0o, φ = 0o), the side-diagonal incidence (θ = 45o,

φ = 0o), and diagonal incidence (θ = 45o, φ = 35o).

7.4.1 Frequency-independent Results

The performance of the investigated boundary model for a wide range of impedance values

is conveniently illustrated in plots for frequency-independent wall impedances, from which

the numerical reflectance for a given frequency and impedance value is clearly visible.

The magnitude of the numerical reflectance for a set of real values of specific acoustic

impedance ξw = 32 , 7

3 , 4, 9, and 10000 is depicted in Figures 7.7 and 7.8. As is the case

for 2D boundaries, the numerical error is in general higher for low impedance values than

for high impedance values. These figures indicate that there are three schemes for which

the theoretical reflection is perfectly matched by the numerical reflectance at normal inci-

dence; these are the octahedral, the tetrahedral, and the interpolated wideband schemes.

Regarding the octahedral boundary model, the cutoff frequencies at two grazing angles of

incidence are low and very close to each other, as illustrated in Figure 7.7(a). Consequently,

a relatively small error for all incidences is ensured for a frequency range up to 0.095fs. As

depicted in Figure 7.7(b), the worst approximation for the tetrahedral boundary results

in diagonal direction and a small error in the numerical reflectance is obtained for up to

0.18fs. The same frequency range for a negligible error in numerical reflectance results

for both isotropic schemes presented in Figures 7.7(c) and 7.8(b). This time the error in

reflectance is most pronounced at normal incidence. The plots of numerical reflectance

cover the whole bandwidth only for the interpolated wideband scheme, which indicates

that the reflectance is provided for all incidences and all frequencies. In particular, the


error in reflectance is kept relatively small for up to 0.19fs. As far as isotropy of the

numerical error is concerned, by far the closest error for all three incidences is observed for

the interpolated digital waveguide mesh, where all reflectance plots look very similar [see

Figure 7.8(a)]. An upper bound on the range of frequencies for which the reflectance error

is independent on the angle of incidence is hard to observe due to a high overall value of

this error. However, a detailed analysis shows that this boundary model is highly isotropic

up to 0.18fs. For the boundaries of both isotropic schemes, the numerical reflectance error

is independent on the angle of incidence for up to 0.27fs.

7.4.2 Frequency-dependent Results

For the frequency-dependent results of 3D compact explicit boundaries, the same 15th-

order filter as for the 2D boundaries is applied. Thus the target response of the filter

representing a layer of fibrous material on a rigid wall is closely matched for frequencies

up to 0.5fs. The theoretical reflectance is again defined as the digital reflectance for a given

impedance filter so that the initial filter design error is eliminated from the comparisons.

The amplitude of the numerical reflectance of the fibrous material layer for a few

special cases of the 3D compact explicit schemes is depicted in Figures 7.9 and 7.10, where

the thickness of fibres is d = 0.06m and the flow resistivity equals σ = 200Nsm−4. In

general, all the presented boundary models follow quite well the theoretical reflectance

up to their respective cutoff frequencies. In particular, the maxima and minima of the

numerical reflectance are positioned correctly for grazing incidences (i.e., the side-diagonal

and diagonal angles of incidence), and the numerical error exhibits itself at high frequencies

in the exaggeration of the variation in reflectance amplitude. The boundary models for

which the numerical error is the strongest at normal-incidence, a slight shift in phase can

also be observed at higher frequencies. Since both isotropic schemes and the interpolated

digital waveguide mesh exhibit such a high numerical error at normal angle of incidence,

these phase shifts make it hard to investigate the numerical error value in amplitude alone.

Furthermore, even theoretical reflectance plot at normal incidence differs substantially

from the other two grazing angles of incidence, which makes it difficult to observe the

isotropy of an error in reflectance in Figures 7.9 and 7.10. However, when investigating

the numerical reflectance error given by Equation (5.63), which takes into account both

amplitude and phase deviations, the resulting frequency ranges for isotropy in numerical

reflectance are very similar to the frequency-independent case. Concerning the accuracy

range in reflectance for various angles of incidence, the interpolated wideband scheme

clearly outperforms the other compact explicit 3D schemes, reaching as far as 0.28fs.

The tetrahedral scheme has a smaller accuracy range than the interpolated wideband

scheme but has a wider frequency range of accurate reflectance than the octahedral scheme.


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1


Re

fle

cta

nce

am

plit

ud

e

θ = 0o, φ = 0

o

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1


Re

fle

cta

nce

am

plit

ud

e

θ = 45o, φ = 0

o

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

1.2


Re

fle

cta

nce

am

plit

ud

e

θ = 45o, φ = 35

o

(a)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1


Re

fle

cta

nce

am

plit

ud

e

θ = 0o, φ = 0

o

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1


Re

fle

cta

nce

am

plit

ud

e

θ = 45o, φ = 0

o

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

1.2


Re

fle

cta

nce

am

plit

ud

e

θ = 45o, φ = 35

o

(b)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1


Re

fle

cta

nce

am

plit

ud

e

θ = 0o, φ = 0

o

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1


Re

fle

cta

nce

am

plit

ud

e

θ = 45o, φ = 0

o

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

1.2


Re

fle

cta

nce

am

plit

ud

e

θ = 45o, φ = 35

o

(c)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1


Re

fle

cta

nce

am

plit

ud

e

θ = 0o, φ = 0

o

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1


Re

fle

cta

nce

am

plit

ud

e

θ = 45o, φ = 0

o

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

1.2


Re

fle

cta

nce

am

plit

ud

e

θ = 45o, φ = 35

o

(d)

Figure 7.7: Numerical reflectance amplitude (dashed black lines) compared to theoretical reflectanceamplitude (solid grey lines) for the following angles of incidence θ = 0o, φ = 0o, θ = 45o, φ = 0o,and θ = 45o, φ = 35o, and real specific acoustic impedances ξw = 3

2, 7

3, 4, 9, 10000 for: (a) the

octahedral boundary, (b) the tetrahedral boundary, (c) the interpolated isotropic 2 boundary, and (d)the interpolated wideband boundary.


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1


Re

fle

cta

nce

am

plit

ud

e

θ = 0o, φ = 0

o

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1


Re

fle

cta

nce

am

plit

ud

e

θ = 45o, φ = 0

o

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

1.2


Re

fle

cta

nce

am

plit

ud

e

θ = 45o, φ = 35

o

(b)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1


Re

fle

cta

nce

am

plit

ud

e

θ = 0o, φ = 0

o

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1


Re

fle

cta

nce

am

plit

ud

e

θ = 45o, φ = 0

o

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

1.2


Re

fle

cta

nce

am

plit

ud

e

θ = 45o, φ = 35

o

(a)

Figure 7.8: Numerical reflectance amplitude (dashed black lines) compared to theoretical reflectanceamplitude (solid grey lines) for the following angles of incidence θ = 0o, φ = 0o, θ = 45o, φ = 0o, andθ = 45o, φ = 35o, and real specific acoustic impedances ξw = 3

2, 7

3, 4, 9, 10000 for: (a) the interpolated

digital waveguide mesh boundary, (b) the interpolated isotropic boundary.

These frequency ranges match quite well the frequency ranges presented for the frequency-

independent results.

7.5 Discussion

The analysis of the presented boundary models for various compact explicit schemes indi-

cates that the error in numerical reflectance manifests itself in a way that is very similar

to that of the dispersion error of the respective schemes. This can of course be expected

when the compact explicit finite difference scheme approximating the wave equation is

used consistently in the room interior and at the boundary. Note however that there is an

additional discretisation error involved at the boundary that stems from the approximation

of the first-order derivatives in Equation (5.4) using centered operators.

Furthermore, the frequency bands of accuracy and isotropy of the presented boundaries

for other angles of incidence always lie between the values resulting for (θ = 0o) and

(θ = 45o) in a 2D case and (θ = 0o, φ = 0o), (θ = 45o, φ = 0o), and (θ = 45o, φ = 35o) for a


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1


Re

fle

cta

nce

am

plit

ud

e

θ = 0o, φ = 0

o

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1


Re

fle

cta

nce

am

plit

ud

e

θ = 45o, φ = 0

o

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

1.2


Re

fle

cta

nce

am

plit

ud

e

θ = 45o, φ = 35

o

(d)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1


Re

fle

cta

nce

am

plit

ud

e

θ = 0o, φ = 0

o

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1


Re

fle

cta

nce

am

plit

ud

e

θ = 45o, φ = 0

o

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

1.2


Re

fle

cta

nce

am

plit

ud

e

θ = 45o, φ = 35

o

(b)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1


Re

fle

cta

nce

am

plit

ud

e

θ = 0o, φ = 0

o

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1


Re

fle

cta

nce

am

plit

ud

e

θ = 45o, φ = 0

o

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

1.2


Re

fle

cta

nce

am

plit

ud

e

θ = 45o, φ = 35

o

(a)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1


Re

fle

cta

nce

am

plit

ud

e

θ = 0o, φ = 0

o

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1


Re

fle

cta

nce

am

plit

ud

e

θ = 45o, φ = 0

o

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

1.2


Re

fle

cta

nce

am

plit

ud

e

θ = 45o, φ = 35

o

(c)

Figure 7.9: Numerical reflectance amplitude of the fibrous material boundary with d = 0.04 and σ = 200(dashed black lines) compared to theoretical reflectance amplitude (solid grey lines) for the followingangles of incidence θ = 0o, φ = 0o, θ = 45o, φ = 0o, and θ = 45o, φ = 35o , for: (a) the octahedralboundary, (b) the tetrahedral boundary, (c) the 3D interpolated isotropic 2 boundary, and (d) the 3Dinterpolated wideband boundary.


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1


Re

fle

cta

nce

am

plit

ud

e

θ = 0o, φ = 0

o

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1


Re

fle

cta

nce

am

plit

ud

e

θ = 45o, φ = 0

o

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

1.2


Re

fle

cta

nce

am

plit

ud

e

θ = 45o, φ = 35

o

(a)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1


Re

fle

cta

nce

am

plit

ud

e

θ = 0o, φ = 0

o

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1


Re

fle

cta

nce

am

plit

ud

e

θ = 45o, φ = 0

o

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

1.2


Re

fle

cta

nce

am

plit

ud

e

θ = 45o, φ = 35

o

(b)

Figure 7.10: Numerical reflectance amplitude of the fibrous material boundary with d = 0.04 andσ = 200 (dashed black lines) compared to theoretical reflectance amplitude (solid grey lines) for thefollowing angles of incidence θ = 0o, φ = 0o, θ = 45o, φ = 0o, and θ = 45o, φ = 35o , for: (a) the 3Dinterpolated digital waveguide mesh boundary, (b) the 3D interpolated isotropic boundary.

3D case, respectively. Therefore, the performance of these boundaries can be generalised

even though results for other angles of incidence are not provided here.

In general, the numerical reflectance of the boundary models proposed in this chapter

for the family of 2D and 3D compact schemes structurally adhere to theory for all angles

of incidence and wall impedances for up to the respective numerical cutoff frequency,

i.e. the proposed boundary models are consistent with locally reacting surface theory.

More specifically, the numerical reflectance is accurate within 2% in 2D and 4% in 3D at

frequencies up to about half the numerical cutoff frequency at a given angle of incidence

for the majority of schemes.

If we define that a boundary is isotropic as long as the difference between the reflectance

errors in axial and diagonal direction is within the range [2−4%], then the results indicate

that the interpolated isotropic boundary is both more accurate and isotropic for a wider

frequency range than the interpolated digital waveguide mesh boundary. However for a

given sample rate, the interpolated wideband boundary is by far the most accurate, uses

the full bandwidth, and also appears to be isotropic for a relatively wide frequency range,


which is almost comparable with the interpolated isotropic boundary.

7.6 Conclusions

In this chapter, a method for modelling boundary reflection in 2D and 3D rooms has been

presented, using compact explicit FDTD schemes. These schemes are based on a rectilinear

mesh, and therefore particularly suited to modelling rooms, which often have parallel walls.

The family of compact explicit schemes includes a number of special cases of interest that

are either overall-accurate or nearly-isotropic, and as such are good candidates for on-line

and off-line auralisation applications, respectively.

Generalised frequency-dependent boundaries are modelled using a compact explicit

digital impedance filter formulation for terminating the compact explicit 2D and 3D mesh.

This formulation is constructed by carefully applying the boundary condition in the di-

rection normal to the modelled wall - which involves interpolation - and combining with

the discrete 2D/3D wave equation to eliminate all ghost points. This approach was also

applied to the treatment of inner and outer corners, which is crucial for ensuring accuracy

and stability of the whole numerical simulation. In addition, a numerical boundary analy-

sis formula is given, which provides an exact evaluation of the boundary reflectance, thus

avoiding the need for running many computationally heavy and time-consuming numerical

experiments.

The presented frequency-independent and frequency-dependent results confirm that

the compact explicit digital impedance filter model is consistent with locally reacting

surface theory and that the numerical reflectance matches the expected theoretical value

very well for a wide range of wall impedances and angles of incidence. The presented

formulation is directly applicable to any scheme that is a member of a more general

family of compact explicit schemes, such as the standard leapfrog and the rotated leapfrog

schemes in a 2D case and the standard leapfrog, the octahedral and the tetrahedral schemes

in the 3D case. This formulation is also suitable for terminating the rectangular and the

interpolated digital waveguide mesh implemented with Kirchhoff variables in 2D and 3D,

respectively. Thus each scheme is now characterised by a boundary model specially tailored

for a particular case, and the respective discrete wave equation consistently applies in the

room and at the boundary.

The consistency of scheme for the room interior and the boundary has clear advantages.

Firstly, the simulation as a whole is consistent with general room acoustics theory (i.e. the

wave equation and locally reacting surface theory). Secondly, the numerical system can

be properly analysed in terms of stability and accuracy. While such consistency may seem

trivial to some readers, one has to bear in mind that many previous discrete-time models

proposed in the literature, in particular those using a digital waveguide mesh (DWM)


with terminating reflection filters, are not consistent with theory due to their boundary

formulation. To the best of author’s knowledge it is therefore the first time that off-line

post-processing frequency warping technique can be justifiably applied in FDTD/DWM

room acoustics simulations.

It has been shown that the interpolated wideband DIF boundary is the most accurate

of all compact explicit digital impedance boundaries, and as the only one provides a

full bandwidth. Therefore taking into account the conclusions of Chapter 4, it can be

summarised that the interpolated wideband scheme is optimally efficient for both room

interior and boundary modelling. Displaying no numerical dispersion in axial propagation

directions and no reflectance error at near-normal incidences is also an excellent property,

since the most pronounced room modes arise from sound wave reflections between parallel

walls; hence simulations using the interpolated wideband scheme are most accurate for

the strongest modes of the system.

186

Chapter 8

A Phase Grating Approach to

Modelling Surface Diffusion

Sound scattering is one of the main acoustic phenomena encountered in room acoustics,

and consists primarily of edge diffraction caused by finite-size boundaries and surface dif-

fusion caused by not perfectly smooth boundaries. This phenomenon is perceptually very

important and thus accurate simulations of room acoustics also require modelling of sound

scattering effects. Whereas edge diffraction is inherently modelled in the FDTD method,

surface diffusion may require special treatment, especially when controllable diffusion is

desired or the surface roughness is small compared to finite difference grid spacing.

In general, simulation tools can significantly help the process of designing spaces with

desirable acoustics, as they allow the acoustic performance to be predicted before the

construction of the building begins. The finite difference time domain method has recently

emerged as a possible numerical tool for investigation of surface diffusion (see, e.g. [127, 84].

In such studies, an irregular boundary geometry is usually represented directly in the mesh

structure, for example a staircase implementation [127, 84]. This works fine for variations

in the boundary shape that are large in comparison to the grid spacing, but adapting to

smaller variations (i.e. subcell geometries) can be problematic, since local grid refinement

leads to a significant increase in computational load [28].

In the context of room auralisation for audio applications, accurate simulation predic-

tivity is in some cases of lesser importance, the main objective shifting to enabling better

control over the overall room diffusivity. For such applications, modelling commercial dif-

fusers is not really suitable since they are usually optimised to provide maximum diffusion

and they do not really allow controlling the strength and frequencies for which scattering

effect occurs.

This chapter addresses modelling diffusion in FDTD simulations by varying the impedance

across the boundary surface, capturing frequency-dependent absorption and diffusion

Chapter 8. A Phase Grating Approach to Modelling Surface Diffusion 187

within one model. Each node in the boundary is represented by an impedance filter

that is designed from a normal-incidence reflection filter consisting of an absorption filter

(that if left by itself would model specular reflections) in cascade with a fractional delay

line. Surface diffusion is realised by variation across the boundary of the delay lengths,

thus imposing a spatial variation of the ‘local well depth’. Hence the proposed approach

is based on the well-known concept of phase-grating diffusers (see Section 2.6 for a brief

overview of this concept).

The chapter is structured as follows. In Section 8.1, a method for simulating phase-

grating diffusers with the use of fractional delay lines is proposed, and ways of affecting

and controlling diffusion are discussed. Results obtained with numerical experiments for

a range of diffusers are analysed in frequency and time domains in Section 8.2, followed

by concluding remarks in Section 8.3.

8.1 A Method for Simulating Diffusive Surfaces

8.1.1 A Phase Grating Approach

A physical approach to modelling diffusion requires addressing the issue of absorption

and diffusion simultaneously. Under the assumption that the surface is locally reacting,

the absorption properties of the boundary can be defined locally. From a physical point

of view, surface diffusion is caused by irregularities in shape along the boundary. Thus,

strictly speaking, they cannot be defined as a property of a single point on the surface.

Hence in order to formulate a model that allows as much as possible to maintain the

natural separation between these two wave phenomena, we propose to model absorption

with local impedance filters, and to model diffusion by varying the impedance along the

boundary surface without changing the local absorption properties. In other words, only

the reflection phase properties are spatially varied. A schematic representation of this

model is depicted in Figure 8.1.

The starting point for the design of each of the impedance filters is to consider first a

locally-defined normal-incidence digital reflection filter of the form

R0(z) = Ra(z) Rd(z), (8.1)

where Ra(z) is a digital reflection filter designed to match the absorption data and Rd

is a fractional delay filter that adds a local ‘well-depth’. From Equation (8.1) it follows

directly that a flat surface that reflects waves in a specular fashion is simulated by setting

Rd(z) = 1 for all impedance filters. Non-smooth surfaces thus result when varying the well

depths (i.e. the delay lengths) along the surface. As mentioned in the introduction, this

may increase the overall surface absorption, even though the local absorption properties


ξ(m) ξ(m+1) ξ(m+2) ξ(m+3)ξ(m-1)ξ(m-2)

(a)

(b)

Figure 8.1: (a) Irregular surface geometry and (b) a way of modelling these irregularities by spatialvariation of digital impedance filters. The impedance filters are designed from normal-incidence filterswith added well-depths, thus implementing a phase grating diffuser.

are not affected by any changes in the Rd(z) filters (i.e. the magnitude response of Rd(z)

filter is flat, as explained in Section 8.1.3).

Once the digital reflection filter is designed, it is readily converted to a digital impedance

filter with Equation (6.1), and incorporated in the boundary model as described in Chap-

ter 6. For any passive boundary (i.e. |R0(z)| ≤ 1), Equation (6.1) yields a positive real

digital impedance filter. As shown in Chapter 6, this also guarantees the overall stability

of the simulation.

8.1.2 Relationship between the Well Depth and Delay Length

For a given wave velocity c, the time it takes to propagate over a well-depth distance d

is τ = d/c. Hence the required fractional delay length [the fractional number of delays

modelled with Rd(z)] is

D =2d

cT. (8.2)

As mentioned in Section 3.2.2, the wave velocity actually differs from c for most frequencies

and directions. For the standard leapfrog scheme, this amounts to no error in diagonal

directions and the largest error occurring at Nyquist for axial directions, where the ratio

between the numerical and real phase velocity becomes√

1/2. Given that it is impossible

to adjust the delay length of a well to all possible directions and frequencies, Equation

(8.2) is the best possible overall design choice. Note that this assumes a normal angle of

incidence, which is the most natural choice since diffuser wells are placed in the direction

perpendicular to the boundary surface. The resulting non-exactness of the delay for certain

directions and frequencies may however lead to small high-frequency deviations in diffusion

characteristics for some diffuser types.


8.1.3 Fractional Delay Implementation

Ideally, the filter Rd(z) should model an exact number of delays, and this number will

generally be fractional (i.e., a non-integer value). The ideal fractional delay element (i.e.,

an ideal band-limited interpolator) is nonrealisable, since the corresponding impulse re-

sponse would be noncausal and infinitely long [118]. Suitable approximations can be made

using so-called fractional delay filters with a maximally flat phase delay response, such as

Lagrange interpolation FIR filters and Thiran allpass filters [66, 118]. The main difference

between these two filter design methods is that the Lagrange FIR filter introduces a low-

pass filtering effect, while the Thiran design is strictly allpass (see Section 3.5 for more

details). Hence the latter were employed in this study, mainly in order to maintain full

control of absorption properties through Ra(z). A simple rule of thumb for the stability

and accuracy of such allpass filters is given as [M − 0.5 ≤ D < M + 0.5], where D denotes

a fractional delay value and M is the filter order [118].

For efficiency reasons, it is often desirable to keep the fractional delay filter Rd(z) of

lower order, such that the final impedance filter also remains small in order; in this study

we therefore employed second-order Thiran allpass filters. Fractional delays larger than

the filter order N + 0.5 can be obtained by combining the allpass filter with a delay line

[106], hence the filter transfer function is

Rd(z) = A(z) z−N , (8.3)

where A(z) denotes the second-order Thiran allpass filter and z−N denotes a delay line

of N = D − 2 delays. For very small fractional delays [0.1 ≤ D ≤ 1.1], a first-order

Thiran allpass filter has to be applied. For D ≤ 0.1, pole-zero cancellation may happen

in practical implementations due to inevitable round-off errors, and should preferably be

avoided, so the minimum fractional delay is set to D = 0.1 [106].

8.1.4 Diffusion Parameter Control

In many practical cases, it is desirable to control the strength of diffusion and the frequen-

cies for which diffusion occurs. For example, controlled diffusion can be used to match

measured diffusion data, to account for small surface roughness or to mask numerical arte-

facts. Modelling commercially available diffusers is not suitable for this purpose since such

optimised surfaces usually yield top values of the diffusion coefficient for the frequencies

of interest. Therefore we propose a technique to control the diffusion properties of the

numerical boundary that is based on fractional Brownian diffusers, which are described in

detail in Section 2.6.6.

This method works as follows. A Gaussian white noise signal is first generated with the


length equal to the number of the mesh nodes along a diffusive surface. Such a noise signal

is then spectrally shaped with the use of a digital filter designed to match values of the

diffusion coefficient in each frequency band. The maximum depth in shape, defined by the

maximum delay value, puts a constraint on the lowest frequencies that are affected by the

diffusion. As will be shown in the results presented in Section 8.2, the lowest frequency at

which diffusion occurs can be estimated in a similar way to defining the theoretical design

frequency of the Schroeder diffuser. The maximum depth dmax should thus be the half of

the longest wavelength

dmax =λ0

2, (8.4)

which also enables computing the lowest frequencies affected by diffusion from

flow =c

2dmax, (8.5)

where flow is the lowest diffusion frequency. However, in practice some slight diffusion

actually starts from frequencies an octave below this theoretical value. For instance,

setting maximum depth dmax = 0.5m results in diffusion from around flow = 150Hz.

The amount of diffusion is obtained by spectral shaping with the use of a simple linear

roll off filter, the gain of which is given as

G(f) =1

fβ/2, (8.6)

where β denotes the spectral density exponent which in order to assure the 1D fractal

shape takes the values within the range of 1 and 3. Low values of the spectral density

exponent yield a more spiky surface shape resulting in a highly diffusive boundary. On the

other hand, for high values of β, the surface shape is much smoother and the diffusion is

less pronounced at frequencies above the design frequency. However, some slight diffusion

may also occur at frequencies around an octave below the design frequency when spectral

density component value is relatively high. Note that high values of the diffusion coefficient

at low frequencies rarely originate from the surface scattering alone, but rather from the

diffraction effects caused by finite-length boundaries. Consequently, this technique should

not be over-used to compensate for such large-size changes. That is, in such cases the

boundary shape should instead be implemented directly in the grid structure.


8.2 Numerical Experiments

8.2.1 Test Setup

The diffusion coefficient characterises the sound reflection from a surface in terms of the

uniformity of the scattered polar distribution, the measurement setup for which is defined

in the AES 4id 2001 standard [3], and described in detail in Section 2.7. The setup used in

the numerical experiments in this study complies with these AES guidelines and is illus-

trated in Figure 8.2. The size of the modelled room (4000x4000 mesh nodes corresponding

to 44m x44m) and the simulation time (4410 samples at the sample rate of 44.1kHz) were

set in such a way that only the reflections from the investigated boundary sample could

reach the receiver positions and at the same time sufficiently long so that the whole wavelet

could reach all the receivers. All the investigated diffusers were implemented using the

fractional DIF boundary model for the 2D standard leapfrog scheme. The diffuser sample

was 101 mesh points wide (which corresponds to 1.1m) and only 3 mesh points deep (which

corresponds to 0.033m). Note that the proposed diffusion method allows for modelling

much deeper wells without having to increase the external depth of the diffuser’s sample.

The sample width was chosen so that at least 6 complete repeat sequences of periodic test

surfaces were included; in most cases over 10 complete sequences resulted. The sample

was also sufficiently large so that surface effects rather than edge diffraction were more

prominent in the scattering. The numerical measurement consisted of 19 simulations, in

which the source position was changed with a maximum angular separation of 10o on a

semicircle with a radius of 7.7m (700 grid points), as shown in Figure 8.2. Similarly, a

set of receivers with an angular receiver resolution of 5o was located on a semicircle with

a radius of 4.4m (400 grid nodes), where sample diffuser is positioned in the semi-circle

centre. Such distances are in a good agreement with the AES standard [3], in which when

true far field conditions are hard to ensure, the minimum criterion is that at least 80% of

the receiver positions should be outside of the specular zone. In addition, the width of the

diffuser places a constraint on the maximum wavelength that is effectively reflected from

the boundary of a finite size. In the case of a 1.1m wide diffuser, the diffusion coefficient

measurement technique can be considered valid for frequencies above this cutoff frequency,

which in this case amounts to 870Hz.

Each numerical test consisted of two simulations, where a sharp impulse was inserted in

the source position. In the first simulation, the receivers’ impulse responses h1 were mea-

sured with the diffuser sample in the middle of the room; h1 inevitably also included the

“direct signal”. In the second simulation, the sample was removed and impulse responses

h2 were measured in the same positions. The isolated reflected signals were obtained as

h = h1 − h2. The virtual microphones and virtual loudspeakers used in the experiments

are treated as omnidirectional since the input and output signals were injected into one


4.4m7.7m

Compuational domain

Figure 8.2: Schematic depiction of the simulation setup according to the AES standard [3]; black squaresindicate omnidirectional source positions and black circles indicate receivers positions on a semicircle.The computational domain is indicated with outer rectangle.

mesh node and picked-up from another mesh node. Hence there was no need to decon-

volve the loudspeaker-microphone response. The impulse response, windowed with the

use of the right half of the Hanning window, was Fourier transformed and sound pres-

sure levels Li (in decibels) were calculated in one-third octave bands. Following [3], the

directional diffusion coefficient was computed in each one-third octave band from the set

of sound pressure levels Li from n receivers for a fixed source position with the formula

given by Equation 2.71. Finally, the random-incidence diffusion coefficient was calculated

as an arithmetic mean of the directional diffusion coefficients for all source positions using

Equation 2.72.

8.2.2 Modelled Diffusers

In this section, some details related to the implementation of the example phase grating

diffusers that were modelled with the use of the proposed technique are provided. An

overview of real diffusers can be found in Section 2.6. Since the focus of this chapter is on

modelling and measuring diffusion, the absorption reflection filter used in Equation (8.1)

is in all cases set to Ra(z) = 1 for simplicity.

Maximum Length Sequence Diffuser

In numerical experiments, the direct change in the boundary reflectance between Ro(z) = 1

and Ro(z) = −1 of a maximum length sequence diffuser was modelled. The length of the


applied MLS was 15 resulting in the following reflection coefficient values (−1, 1, 1,−1, 1,

−1, 1, 1, 1, 1,−1,−1,−1, 1,−1), and such a periodic sequence was repeated almost 7 times.

Quadratic Residue Diffuser

A diffuser based on the 7-ary quadratic residue sequence was modelled, for which the

periodic sequence (0, 1, 4, 2, 2, 4, 1) is shown in Figure 8.3(a). The QRD sequence was

repeated over 14 times and the respective delays in the delay filter Rd(z) were obtained

by multiplying each sequence value by 3, which resulted in the following delay values

(0, 3, 12, 6, 6, 12, 3). The number 3 is chosen arbitrarily so that the maximum delay of

the 7-ary QRD amounts to 21 sample delay, which corresponds to the maximum well

depth dmax = 0.163m and the design frequency fo = 2100Hz, respectively. Note that

when manufacturing this type of diffusers, a −1 reflection is typically realised by a quarter

design wavelength deep grooves in hard walls; however, a direct change in impedance is

difficult to achieve in practice. In the proposed FDTD simulation method, the realisation

of an MLS diffuser is trivial, by simply adjusting the local impedance filters.

Modulated QRD

In the modulated phase reflection grating technique, the inverted 7-ary quadratic residue

sequence was (7, 6, 3, 5, 5, 3, 6), as depicted in Figure 8.3(b), and the MLS of a length

15 (0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0) was applied as the pseudorandom binary sequence.

Similarly, the basic and inverted sequence values were multiplied by 3, giving the values of

delays in samples which were applied in delay filters. Secondly, two QRD diffusers based

on the 7-ary sequence and 5-ary sequence were orthogonally modulated depending on the

value of the binary MLS sequence of length 15. Two successive sequences, illustrated in

Figure 8.3(c), are given by [7]

dn =

(0,

1

7,4

7,2

7,2

7,4

7,1

7, 0,

1

5,4

5,4

5,1

5

)dmax, (8.7)

where the maximum depth dmax = 0.163m corresponds to a 21 sample delay.

Fractional Brownian Diffusers

The numerical simulations of the Fractional Brownian Diffusers were conducted for three

sequences, namely a white noise and two Brownian noise sequences, the latter resulting

from spectral shaping of a white noise with the spectral density exponent values β = 1.73

and β = 3, respectively. These three noise sequences are illustrated in Figure 8.4. In

successive experiments, the maximum delay values corresponding to the maximum depth

of the diffuser, were set to D = (2, 5, 10, 20) samples for all three diffusers.


0

2

4

1

3

7 sequence 7 sequence

0 01 1 1 14 4 4 42 2 2 2

0

2

4

1

3

inverted 7 sequence

5

7

6

7 sequence

0 11 4 4 7 662 2 3 35 5

0

2

4

1

3

5

7

6

0

2

4

1

3

5

7 sequence 5 sequence

0 1 4 01 14 14 42 2

(c)

(b)

(a)

Figure 8.3: The well depths for the following quadratic residue diffusers: (a) 7 sequence, (b) modulatedQRD with 7 sequence, and (c) orthogonal QRD with 7 and 5 sequences.

8.2.3 Frequency-domain Results

In this section, the diffusive properties of the aforementioned diffusers are analysed in the

frequency domain. Spatial diffusion as a function of frequency is shown in magnitude

pressure plots for a set of receivers, whereas the overall diffusivity is depicted in figures

presenting the random-incidence diffusion coefficient as a function of frequency.

The magnitude pressure results for a set of receivers and a normal angle of incidence

(i.e., source position at θ = 0o) for a flat surface, an MLS diffuser, and a set of quadratic

residue diffusers are illustrated in Figure 8.5(a-e). The pressure amplitude of the sound

wave reflected from the flat panel sample is illustrated in Figure 8.5(a), and its diffusion

coefficient is shown in Figure 8.6. The diffusion coefficient for a flat surface is in a very

good agreement with measured data provided in [23], and is used here as a reference for

the investigated diffusers.


(a)

(b)

(c)

Figure 8.4: Boundary surface shapes: (a) white Gaussian noise, (b) Brownian noise with β = 1.73, and(c) Brownian noise with β = 3.

Firstly, the maximum length sequence was implemented as an example of varying

impedance along the boundary instead of changing well depths. One obvious implication

of such an implementation is that a diffuser should cause scattering for all frequencies due

to the lack of the design frequency. The characteristic diffuser lobes are well pronounced

in Figure 8.5(b), and the diffusion coefficient is presented in Figure 8.6. These figures

indicate that even though some diffusion is actually obtained for all frequencies, high

diffusivity results for frequencies above fo = 2.2kHz. At lower frequencies the diffusion

is much smaller probably because of the fact that low-frequency waves tend to average

across rapidly changing local impedance values, thus effectively diminishing the scattering

effect.

The pressure magnitude for the quadratic residue diffuser of a sequence length 7,

the modulated version of such a diffuser, and the orthogonal QRD for sequences 7 and 5,

according to the binary MLS sequence of the length 15, are presented in Figures 8.5(c),(d),

and (e), respectively. Since there is very small variation in the depths of the quadratic

residue diffuser of a length 7 and such a quadratic residue sequence has been repeated over

the whole boundary sample 14 times, the resulting diffusion coefficient is relatively small,

as illustrated in Figure 8.7. Furthermore, the diffusion is not yet effectively observed at the

design frequency fo = 2100Hz, even though it slowly begins to rise from this frequency. On

the other hand, the modulated quadratic residue diffuser exhibits much higher diffusion

coefficient values, which can also be observed in the pressure amplitude plot depicted in

Figure 8.5(d). This effect is due to the greater variation in the well depths and the lack of

obvious repetition in the boundary shape pattern. A significantly increased diffusivity at


(e)

Frequency [kHz]

Reveiv

er

angle

[degre

es]

(d)

Frequency [kHz]

Reveiv

er

angle

[degre

es]

(c)

Frequency [kHz]

Reveiv

er

angle

[degre

es]

(b)

Frequency [kHz]

Reveiv

er

angle

[degre

es]

(a)

Frequency [kHz]

Reveiv

er

angle

[degre

es]

Figure 8.5: The pressure magnitude of the reflected signal for all receiver positions located on asemicircle for: (a) a flat surface, (b) a maximum length sequence, (c) a QRD sequence 7, (d) amodulated QRD sequence 7, and (e) an orthogonal QRD sequences 7 and 5.


200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000 6300 80000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency [Hz]

Diff

usio

n co

effic

ient

MLSFlat surface

Figure 8.6: Random-incidence diffusion coefficient for a flat surface (black dotted line) and a maximumlength sequence diffuser (black solid line).

200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000 6300 80000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency [Hz]

Diff

usio

n co

effic

ient

QRDModulated QRDOrthogonal QRD

Figure 8.7: Random-incidence diffusion coefficient for a quadratic residue diffuser of length 7 (blacksolid line), modulated 7 sequence QRD (grey solid line), and an orthogonal QRD for sequences 7 and5 (black dashed line).

low frequencies is an obvious implication of increasing the maximum well depth which is

used in the inverted QRD. As a result, a high scattering in Figure 8.5(d) and large diffusion

coefficient value is observed for the design frequency. Similarly, the orthogonal diffuser

demonstrates yet another method of increasing the diffusivity of concatenated QRDs, for

which the pressure magnitude is depicted in Figure 8.5(e) and the diffusion coefficient is

illustrated in Figure 8.7. The resulting diffusion is higher than for a single sequence and

a modulated quadratic residue diffuser at the design frequency, and additionally has more

lobes in the pressure magnitude plots. However, the diffusion coefficient values above the


design fequency are lower than for the modulated QRD which can be explained by the

smaller number of available well depths; in particular, the lack of the deepest well. Both

latter cases indicate how important modulation is for quadratic residue diffusers when

multiple concatenated identical shape patterns are applied to cover large wall areas.

The random-incidence diffusion coefficient for a white noise diffusion boundary for

four maximum delay values Dmax = 2, 5, 10, 20 samples is illustrated in Figure 8.9. From

the combination of Equations (8.5) and (8.5), these delays correspond to the following

design frequencies fo = 22.050kHz, 11.025kHz, 4.410kHz, 2.205kHz. In general, the diffu-

sion effect of the white noise Brownian diffuser gradually increases starting from around

two-thirds octave below the design frequency and peaks at one-third octave above the

design frequency, and next remains constant. The top value of the diffusion coefficient is

very similar for all white noises, and the lower the design frequency (i.e., longer maximum

delay) the steeper the increase slope is. For instance, gradual increase in diffusion for

fo = 22.050kHz actually starts much earlier (i.e., over an octave below the design fre-

quency). The diffusion effect is clearly visible in the pressure magnitude plots for a set of

semicircular receivers depicted in Figure 8.8(a-d), which is a much more indicative figure

than just simple polar plots for one frequency as it enables the observation of the diffu-

sive boundary performance for many frequencies simultaneously. Figure 8.8(a-d) shows

that the specular reflection is very well scattered by a white noise signal boundary for

frequencies above the respective design frequencies, and that some slight diffusion occurs

also within an octave below the design frequency. This figure as well as Figure 8.9 also

indicate that the white noise shaped boundary generally exhibits a very high diffusivity.

In order to control the amount of diffusion, spectral shaping of the input noise signal

can be applied. The random-incidence diffusion coefficient for an example design frequency

fo = 4.410kHz is depicted in Figure 8.10 for three noise signals. For higher values of the

spectral density exponent, the diffusion coefficient is substantially smaller than for the

Gaussian noise for frequencies above the design frequency. However, this comes at the

expense of a slightly increased diffusion for frequencies below the design frequency too.

Furthermore, the directional scattering of these diffusers can be investigated in Figure

8.8(c,e,f), which illustrates the spatial distribution of pressure magnitudes at receiver

positions for the same three noise signals. Contrary to the Gaussian noise boundary for

which the incident sound is scattered evenly in many directions [as illustrated in Figure

8.8(c)], the Brownian noise with β = 1.73 also has a strong specularly reflected component

for frequencies above the design frequency fo = 4.410kHz, which is clearly visible up to

around fo = 5.5kHz. Above this frequency the reflected incident wave is scattered quite

randomly in the same fashion as for the white noise boundary, which explains why the

diffusion coefficient increases quite substantially in Figure 8.10 for β = 1.73.

As depicted in Figure 8.8(f), an incident sound is scattered in a few reflection directions


(e)

Frequency [kHz]

Reveiv

er

angle

[degre

es]

(d)

Frequency [kHz]

Reveiv

er

angle

[degre

es]

(c)

Frequency [kHz]

Reveiv

er

angle

[degre

es]

(b)

Frequency [kHz]

Reveiv

er

angle

[degre

es]

(a)

Frequency [kHz]

Reveiv

er

angle

[degre

es]

(f)

Frequency [kHz]

Reveiv

er

angle

[degre

es]

Figure 8.8: The pressure magnitude of the reflected signal for all receiver positions located on asemicircle for the following boundary surface shape: (a) white noise with Dmax = 2 samples, (b)white noise with Dmax = 5 samples, (c) white noise with Dmax = 10 samples, (d) white noise withDmax = 20 samples, (e) Brownian noise with β = 1.73 and Dmax = 10 samples, (f) Brownian noisewith β = 3 and Dmax = 10 samples.


200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000 6300 80000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency [Hz]

Diff

usio

n co

effic

ient

White noise (max 2 samples)White noise (max 5 samples)White noise (max 10 samples)White noise (max 20 samples)

Figure 8.9: Random-incidence diffusion coefficient for a white-noise like boundary shape with themaximum roughness expressed in terms of the number of delay samples Dmax = 2, 5, 10, 20 samples.

200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000 6300 80000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency [Hz]

Diff

usio

n co

effic

ient

White noiseBrownian noise (β=1.73)Brownian noise (β=3)

Figure 8.10: Random-incidence diffusion coefficient for a white-noise (black solid line), Brownian noisewith β = 1.73 (grey solid line), and Brownian noise with β = 3 (black dashed line); the maximumroughness of 10 delay samples.

only for β = 3. Therefore, it can be concluded that for high values of the spectral

density component, the slow variation in the boundary shape redirects an incident sound

rather than scatters it in multiple directions. It can also be observed that since the

boundary shape is based on a random noise sequence, the scattering is neither periodical

nor symmetric, which is for instance illustrated in a polar plot at four example frequencies

depicted in Figure 8.11.


0o

60o

90o

−20dB 0dB

30o

−30o

−60o

−90o

−60dB −40dB

0o

60o

90o

−20dB 0dB

30o

−30o

−60o

−90o

−60dB −40dB

0o

60o

90o

−20dB 0dB

30o

−30o

−60o

−90o

−60dB −40dB

f=2.5kHz f=5kHz

f=7.5kHz f=10kHz

0o

60o

90o

−20dB 0dB

30o

−30o

−60o

−90o

−60dB −40dB

Figure 8.11: Polar magnitude plot for the Brownian noise-like boundary sample with the maximumdepth corresponding to 10 delays and β = 3, at frequencies f = 2.5kHz, 5kHz, 7.5kHz, and 10kHz.

8.2.4 Time-domain Results

The FDTD method is suitable for direct investigation of the diffusive properties of the

modelled diffusers in the time domain. Sound dispersion in time is of great importance

to the overall diffusion effect, the influence of which may even be compared to the spatial

sound redistribution [21]. Research into the time spreading aspect of sound scattering

can be found in some previous studies (e.g. in [84]), especially that this effect plays a

significant part in all phase grating diffusers. It is important to realise that such time

spreading properties are not taken into account when calculating the diffusion coefficient,

and therefore such a time domain analysis is a complementary means of analyzing the

performance of phase grating diffusers. Each time domain plot presented in this section

includes a reflected sound signal from the flat boundary sample (black dashed line) made

of a highly reflective material for comparison.

Figure 8.12(a) and (b) illustrates the reflected signal from the quadratic residue diffuser

sequence 7 and the orthogonally modulated QRD consisting of sequences 7 and 5, for which

time spreading and a reduction in reflected signal amplitude are clearly visible. The

comparison of these results indicates that comparable dispersion occurs for both these

diffusers, even though the latter one has significantly higher diffusion coefficient values.

The signal reflected from a Gaussian-noise shape boundary for fo = 22.050kHz and

fo = 4.410kHz is illustrated in Figure 8.12(c) and (d), respectively. The comparison


Time [samples]1600 1700 1800 1900 2000

−6

−3

0

3

6x 10

−3

1600 1700 1800 1900 2000−6

−3

0

3

6x 10

−3

1600 1700 1800 1900 2000−6

−3

0

3

6x 10

−3

1600 1700 1800 1900 2000−6

−3

0

3

6x 10

−3

(a)

Time [samples]

Am

plit

ud

e

(b)

Time [samples]

Am

plit

ud

e

(c)

Time [samples]

Am

plit

ud

e

(d)

Am

plit

ud

e

(e)

Time [samples]

Am

plit

ud

e

1600 1700 1800 1900 2000−6

−3

0

3

6x 10

−3

Figure 8.12: The signal reflected from sample diffuser (grey solid line) plotted against the signal reflectedfrom the flat boundary sample (black dashed line) in the time domain for the following diffusers: (a)a quadratic residue diffuser of sequence 7, (b) an orthogonal quadratic residue diffuser consisting ofsequences 7 and 5, (c) a white-noise like boundary surface shape with the maximum depth correspondingto 2 delays, (d) a white-noise like boundary surface shape with the maximum depth corresponding to10 delays, (e) a Brownian-noise like boundary surface shape with the maximum depth corresponding to10 delays and β = 3.


of these two plots indicates that for the same ratio in boundary shape variation, more

time spreading occurs for a larger maximum depth of such variations. Nevertheless, the

comparison of a white-noise shape boundary [Figure 8.12(d)] with a Brownian-noise shape

boundary for β = 3 shown in Figure 8.12(e) for the same design frequency fo = 4.410kHz,

indicates that the time spreading effect is also dependent on the boundary shape. A more

spiky shape causes an increased time dispersion.

It can be concluded that in general, diffusers characterised by a strong spatial scatter-

ing (measured as diffusion coefficient) also exhibit strong time spreading. The dispersion

is primarily dependent on the maximum depth of the wells or in other words the maxi-

mum variation in the boundary shape. Furthermore, the ratio of such a boundary shape

variation also plays an important role in the time domain diffusion.

8.3 Conclusions

In this chapter, a novel method for modelling surface diffusion in FDTD room acoustic

simulations is proposed. The main advantage of the proposed method is that diffusion

and absorption can be modelled simultaneously in a manner that is consistent with the

physics of real-life boundary surfaces, and are invariant to the type of acoustic enclosure.

That is, diffusion is controlled only by shaping the geometry of the boundary (thus not

defined locally as in geometrical methods), while the absorption can be locally specified

in an arbitrary frequency-dependent way. To the best of author’s knowledge, this is not

possible with any methods presented in prior studies.

Consequently, FDTD simulation can now be more reliably applied by acoustic engineers

to predict the acoustics of rooms and verify the performance of commercially available

phase grating diffusers. The proposed method also lends itself well to simulating surface

roughness, such the case for example in panels and unplastered walls. We remind the

reader here that sparse changes in the boundary shape are much better modelled by

direct incorporation in the grid. In this sense, the proposed model is complementary to

earlier work on numerical modelling of diffusive boundaries [84, 127].

The results presented in this chapter are generally in good agreement with the liter-

ature, thus validating the methodology. The spatial scattering properties are shown to

depend primarily on the ratio of variation in the boundary shape and the maximum depth

of such changes. Furthermore, the maximum well depth defines the lowest frequency at

which scattering effectively occurs. Hence just as with real-life phase grating diffusers,

large well-depths are necessary in the proposed method in order to obtain low frequency

diffusion. In addition, time spreading properties of various diffusers have been investigated.

Similarly to spatial scattering, surface-induced dispersion has been shown to depend on

the maximum depth of the boundary wells and how smoothly the well depth varies across


the surface (for Brownian diffusers this property is controlled through the spectral density

factor). Interestingly, a high value of the diffusion coefficient does not guarantee strong

time spreading.

Finally, the results indicate that spectral shaping of Gaussian noise is a promising

technique for controlling the diffusive properties of the modelled surface. The results

indicate the potential to simulate scattering behavior that roughly matches a given set

of diffusion coefficient data. Indeed, finding a precise way of mapping diffusion data to

Brownian diffuser model parameters would be an interesting area of future research.

205

Chapter 9

Conclusions and

Recommendations

9.1 General Conclusions

This thesis presents a complete method for modelling room acoustics using the finite

difference time domain technique which is applicable to audio and architectural design

applications. The main aim of this research has been to improve the accuracy of room

acoustics modelling methods by investigating high accuracy FDTD schemes, improved

boundary models of locally reacting surfaces, and methods for modelling controllable sur-

face diffusion. These goals have been successfully achieved in this thesis. The proposed

methods can be applied in improved simulations of naturally sounding immersive sound

environments, spatial sound effects for multimedia and architectural design applications.

The methods presented in this thesis for simulating room acoustics are also directly ap-

plicable to modelling other acoustic systems based on solving the wave equation, such as

membranes.

With regard to modelling the medium, a wide range of compact explicit and implicit

FDTD schemes based on unstaggered rectilinear grids has been investigated, and the

most accurate and isotropic schemes have been identified. Among explicit schemes, a new

scheme that provides the full bandwidth in 2D and 3D simulations has been identified,

which is particularly meaningful in a digital audio context. This scheme, referred to here

as the interpolated wideband (IWB) scheme, has been shown to be optimally efficient

among compact explicit schemes for 2D simulations, regardless of the error criterion.

In the 3D case, the IWB scheme is comparable in efficiency to the isotropic (IISO and

IISO2) schemes, where the latter become more efficient for error criteria smaller than 4%.

However, the isotropic schemes do not have the property of a full simulation bandwidth

in all directions. Alternating direction implicit methods were also considered, and have

Chapter 9. Conclusions and Recommendations 206

been shown even more efficient for a tight error criterion. However, such implicit methods

were not further pursued due to the massive complexity of formulating boundaries for

generally irregular domains in a manner that is consistent with the room interior scheme.

Consequently, the IWB scheme is recommended as the best scheme for updating the room

interior nodes for most applications.

As far as boundaries are concerned, a new method for constructing numerical formula-

tions of locally reacting surfaces for FDTD schemes based on unstaggered grids has been

proposed. Novel formulations for frequency-independent terminations, simple frequency-

dependent walls (incorporating linear resistance, inertia and restoring forces), and gener-

ally frequency-dependent boundaries have been proposed for all compact explicit FDTD

schemes. These formulations have been derived by carefully constructing the boundary

conditions and combining them with the respective discrete wave equations. Such com-

putationally efficient scheme-consistent formulations have been explicitly provided for all

compact explicit schemes. The presented frequency-independent and frequency-dependent

results confirm the high accuracy of the proposed boundary formulations, the reflectance

of which closely matches locally reacting surface theory. In particular, the theoretical

reflectance magnitude is very well matched for all angles of incidence and wall impedance

values. These results clearly indicate that the proposed accurate and isotropic schemes

are also very accurate in terms of numerical boundary reflectance, and outperform directly

related methods such as Yee’s scheme and the standard digital waveguide mesh.

The proposed boundary formulations include the full treatment of inner and outer

corners and boundary edges, which are crucial for stability and accuracy of the whole

simulation. Interestingly, this problem seems to have never been addressed in the literature

in the context of FDTD/DWM simulations of room acoustics.

The proposed boundary models constitute a significant improvement over the com-

monly used 1D boundary model, which in the frequency-dependent case combines FDTD

room interior implementation with reflectance filters at boundaries using KW-converters.

The presented results indicate that the reflectance of such 1D models differs wildly from

theory and may even lead to instability problems, which indicates that their use in ter-

minating multidimensional simulations should be avoided. Consequently, the boundary

models proposed in this thesis should always be applied to terminate Kirchhoff variable

digital waveguide mesh and FDTD room acoustics simulations based on nonstaggered

grids.

In comparison to the simple frequency-dependent boundary model formulated by Bot-

teldooren for staggered grids, the proposed models do not introduce an additional stability

bound allowing for a completely free choice of physically feasible impedances. Furthermore,

the proposed digital impedance filter boundary model outperforms the Botteldooren’s for-

mulation as it enables general frequency-dependency.


In addition, a new analytic method for the exact prediction of the numerical boundary

reflectance has been introduced. This numerical boundary analysis method has been

experimentally confirmed as a precise tool for analytic evaluation of multi-dimensional

boundaries, thus avoiding the necessity for running time-consuming and computationally

heavy numerical experiments. Such NBA formulae have been provided for all the proposed

boundary models and used to prove the overall stability of the simulation.

Furthermore, the digital impedance boundary model can be extended to model con-

trollable diffusion. A method for modelling surface diffusion has been proposed, in which

the diffusion and absorption are simultaneously modelled in a way that is consistent with

the physics of real-life boundaries. The diffusion is controlled by shaping the geometry of

the boundary surface, while absorption is specified locally by the value of the boundary

impedance. This technique is suitable for modelling phase grating diffusers with infinitely

thin separators and modelling high-frequency diffusion caused by small roughness. The

latter is particularly useful for auralisation purposes as it also helps to overcome possible

audible artifacts which stem from discretisation. In order to model diffusion of irregular in

shape surfaces and to create controllable diffusion for auralisation purposes, spectral shap-

ing of the white noise is proposed, thus implementing fractional Brownian diffusers. The

presented results indicate the potential of this approach to model diffusion that roughly

matches a set of diffusion coefficient data.

Finally, some concluding remarks on a complete method for modelling 2D/3D room

acoustics can be made. The importance of consistency in scheme for the room interior

and the boundary should be stressed as a crucial factor for analytically predictable and

stable simulations. For that reason, the combination of the alternating direction implicit

technique with explicit boundary formulations for modelling acoustics of complex-shaped

rooms is not recommended since the overall stability of such a simulation cannot be guar-

anteed. However, the applicability of the ADI implementation to modelling membranes

with clamped edges is encouraged as it yields the best accuracy.

With regard to off-line simulations, the choice between the interpolated digital waveg-

uide mesh and the isotropic scheme depends on the criterion associated with the appli-

cation. It should be emphasized that due to scheme-consistent boundary formulations

provided in this thesis, to the best of author’s knowledge it is the first time that frequency

warping technique can be justifiably applied to FDTD/K-DWM simulations.

For on-line simulations with moving sources and receivers, the interpolated wideband

scheme is recommended for simulations of room acoustics. Apart from being an excellent

choice regarding accuracy and efficiency, it provides a full bandwidth and exhibits no error

in axial room modes that arise from sound wave reflections between parallel walls. From

the perceptual point of view, the advantage of providing full bandwidth might well make

the interpolated wideband scheme generally the best choice for auralisation, regardless of


the accuracy criterion and whether the application is on-line or off-line.

9.2 Future Challenges

One interesting topic for future research is to formulate the digital impedance filter model

for compact implicit schemes, which would provide means for highly accurate simulations

with scheme-consistent implicit approach. However, such a model would additionally

bring about the necessity to apply efficient algorithms for 2D or 3D matrix inversion,

other than the ADI technique. The wide body of knowledge on such multidimensional

matrix inversion methods is available and such research should focus on most efficient

techniques that take into account a generally complex shape of the modelled space.

Whereas such a high accuracy could be very beneficial for architectural design appli-

cations, its advantage to virtual acoustics applications is not that straightforward. This

leads to another research topic in the area of FDTD room acoustics, which involves eval-

uating the perceptual importance of the dispersion error. Interesting research questions

in this context include defining the maximum value of the dispersion error which can

be considered perceptually insignificant. Such research should also lead to defining the

threshold for the maximum admissible difference between dispersion in various directions

and frequency bands. Even though thresholds used in this thesis can be to some extent

indicative, they are not scientifically confirmed and require further research.

Apart from modelling locally reacting boundaries, research into modelling walls of a

nonlocally reacting surface type should be pursued. Nonlocally reacting surface boundary

models might be of great interest, especially when simulating the acoustics of concert

halls with sound reinforcement systems in which low frequencies are dominant. Since real

low-frequency acoustic waves travel along walls to higher floors, this effect should also be

captured by predictive software.

As far as modelling diffusion is concerned, an interesting topic for future research

would be to find a systematic way to map diffusion coefficient data to Brownian diffuser

parameters. This would enable creating surface diffusion that precisely reflects measured

data when the information about the shape of the boundary is unavailable.

Finally, the issue of computational efficiency should be addressed. The computational

load of FDTD methods is high, and thus a real-time implementation is not straightfor-

ward. One possible solution to the computational efficiency problem is to use hybrids

combining the FDTD method at low frequencies and geometrical methods at high fre-

quencies. Another approach would be to develop strategies for exploiting the inherent

parallelism of the FDTD technique. It is envisaged that real-time or near real-time imple-

mentation could be achieved with the use of programmable hardware such as clusters of

FPGA (Field Programmable Gate Array) or multicore programmable architectures such


as graphical processing units.

210

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