Acoustic holography LMS Test

Acoustic holography

LMS Test.Lab

Rev 12A

Copyright LMS International 2012

Rev 12A 3

Table of Contents

Chapter 1 Introduction ................................................................................................... 5

Chapter 2 Acoustic holography concepts .................................................................... 7

Section 2.1 Temporal and spatial frequency ........................................................... 7 Section 2.2 Time domain ........................................................................................ 8 Section 2.3 Spatial domain ..................................................................................... 8 Section 2.4 (Back) propagating to other planes .................................................... 12 Section 2.5 The Wiener filter and the AdHoc window ......................................... 13 Section 2.6 Derivation of other acoustic quantities .............................................. 14

Chapter 1 Introduction

Rev 12A 5

Chapter 1 Introduction

Acoustic holography allows you to accurately localize noise sources. It

therefore helps in both the reduction of unwanted vibro-acoustic noise and

optimization of noise levels.

It :

estimates the acoustic power and the spectral content emitted by the object

under examination.

maps sound pressure, velocity and intensity on the measurement plane and

on all parallel planes. The mapping of these acoustical quantities outside

the measurement plane is done through acoustical holography (near field -

far field).

estimates the acoustic level of the principal sources, including contribution

analysis.

This document describes the principles of taking acoustic measurements and the

subsequent analysis of acoustic holography data, for both stationary and

transient measurements.

Basic principles

In performing acoustic holography, you need to measure cross spectra between

a set of reference transducers and the hologram microphones. From these

measurements you can derive sound intensity, particle velocity and sound

power values.

A basic assumption is that you are operating in free field conditions and that the

energy flow is coming directly from the source. Measurements need to be taken

close to the source.

It provides you with an accurate 3D characterization of the sound field and the

source with a higher spatial resolution than is possible with conventional

intensity measurements.

Chapter 2 Acoustic holography concepts

Rev 12A 7


In This Chapter

Temporal and spatial frequency ......................................... 7

Time domain....................................................................... 8

Spatial domain .................................................................... 8

(Back) propagating to other planes..................................... 12

The Wiener filter and the AdHoc window ......................... 13

Derivation of other acoustic quantities ............................... 14

The principle of acoustic holography is to decompose the measured pressure

field in plane waves, by using a spatial Fourier transform. With the frequency

being fixed, we can calculate how each of these plane waves propagates, and by

adding them we can find the pressure field on any plane which is parallel to the

measurement plane.

Consider an acoustic wave. Measuring the pressure on a plane means cutting the

wavefronts by the measurement plane:

The goal is to determine the whole acoustic wavefront from the known pressure

on the measurement plane. Each microphone in the array measures the

complex pressure (amplitude and phase).

Section 2.1 Temporal and spatial frequency

In considering how to do this we will compare the time and the spatial domain.


8 LMS Test.Lab Acoustic holography

Section 2.2 Time domain

When considering measurements in the time domain, then the position from the

sound source (m) is fixed and we obtain a measure of the pressure variation as a

function of time.

The transformation from the time to the frequency domain is achieved using the

Fourier Transform given below

Section 2.3 Spatial domain

If we now consider measurements where time is fixed and pressure varies as a

function of distance, we can obtain a measure of energy flow.

The spatial frequency of this function or wavenumber (k0) is defined as :

If we fix the temporal frequency, this means that the acoustic wavelength is

fixed too.


Rev 12A 9

The complex pressure as a function of the space is called the pressure image at

the specified frequency.

Conversion from the spatial domain is also done using a Fourier transform. In

Acoustic holography pressure is measured in two dimensions (x and y for

example), so a 2-dimensional transformation is performed.

where S (kx , ky) is the spatial transform of the measured pressure field to the

wavenumber (kx and ky)

domain resulting in the 2-D hologram pressure field.

A measured pressure (sound) wave with a particular temporal frequency can

propagate in a number of directions, so the wavenumber vector (k) will have a

number of components. The appearance of these vectors depends on the plane

on which you are looking at them. The aim is to find the components of these

vectors in the 2 dimensions that define the plane and to do this projections of

the vectors in the plane are made.

Summation of plane waves



The spatial Fourier transform implies that a measured pressure field can be

considered as a sum of sinusoidal functions.

Each of these sinusoidal functions can be understood as the result of cutting the

wavefronts of a plane wave by the measurement plane.

There is a coincidence between the nodes of the sinusoidal function and the

wavefronts. In effect, decomposing the pressure field into a sum of sinusoidal

functions means decomposing the real acoustic wave into a sum of plane waves.

Whatever the angle of incidence, the spatial periodicity must be greater than the

wavelength (l). Propagating and evanescent waves

There are two kinds of plane waves :

propagating waves - whose level remains the same as they propagate but

who undergo a phase shift.

evanescent waves - whose level decreases as they propagate.

Propagating waves represent the sound field that is propagated away from the

near towards the far field. Evanescent waves describe the complex sound field


Rev 12A 11

in the near field of the source.

To understand why we must take evanescent plane waves into account, let us

consider our decomposition of the pressure field into sinusoidal functions. If the

spatial periodicity of a sinusoidal function is shorter than the wavelength, it

cannot be the result of cutting a propagating plane wave by the measurement

plane :

Whatever the direction of the propagating plane wave may be, there is no

possible coincidence between the nodes of the sinusoidal function and the

wavefronts. Therefore, this sinusoidal function must be understood as the

intersection between an evanescent wave (which can have a smaller spatial

periodicity than propagating waves) and the measurement plane.

A mathematical interpretation of the evanescent waves is based on the value of

kz which is the component perpendicular to the measurement directions in the

wave number domain.

kz can be determined from the wave number k0 and the known values of kx and

ky from the transformation.



kz is real when (the spatial periodicity is greater than

the wavelength). This means that the waves lie in the circle defined by the

radius w/c in the wave number domain. kz is imaginary outside of this region.

When kz is imaginary, the propagation factor becomes a damped exponential

function (e-jkz z) meaning that a propagated wave undergoes an amplitude

modification while the phase is not changed.

Section 2.4 (Back) propagating to other planes

Pressure levels at other planes can be found using Raleigh's integral Equation

with Dirichlet's Green function :

where the Green function Gd can be thought of as the transformation function to

transform the sound pressure field from one plane to another. We can use wave

domain properties (k) to predict the pressure at a different spatial position (z).

The practical computation of Raleigh's equation is:

where z' is the measurement plane and z is the position of the required plane.

The green function is given by

The final step is to perform an inverse transformation back to the temporal

domain.


Rev 12A 13

Section 2.5 The Wiener filter and the AdHoc window

As mentioned above, evanescent waves undergo a change in amplitude when

propagating. Propagating towards the source implies an amplification of the

signal that is a function of kz. Evanescent waves that lie far away from the unit

circle have a large kz, therefore their amplitude is amplified significantly when

propagating to the source. The contribution of these evanescent waves results

in an increase of spatial resolution. Note that the inclusion of evanescent waves

is only appropriate when propagating towards the source.

Propagating away from the source, the evanescent waves decrease so rapidly in

amplitude that their contribution to the spatial resolution becomes negligible.

However the further away a wave is located from the circle, the less accurate

the amplitude estimate becomes so that at a certain point noise is propagated

and at that point the propagated image starts to blur.

When propagating towards the source, a Wiener filter can be used to include a

certain number of evanescent waves to improve the resolution. Taking a higher

number of waves taken into account may result in the amplification becoming

unstable. This depends on a parameter of the Wiener filter known as the Signal

to Noise Ratio (SNR). When the SNR value is greater than 15dB, then the

amplification will become unstable as the number of evanescent waves included

increases. Using an low SNR value (5dB for example) means that the

evanescent waves are taken into account but they are so attenuated that the

improvement in resolution is negligible. The default value of 15dB provides the

best compromise in terms of resolution and amplification.

When the Wiener filter is used, the pressure image needs to be multiplied by a

two-dimensional window. As is the case with a single FFT, the observed

pressure must be 'periodic' within the observed hologram. If this is not the case,

then truncation errors occur as with a single FFT. These truncation errors

manifest themselves as ghost sources at the borders of the observed area.

Two windows are used:

The rectangular window,

which does not modify the pressure image. In case of a rectangular window,

only propagating waves are included in the calculations resulting in a resolution

equivalent to an intensity measurement.

The so-called Ad Hoc window

For a time signal, the FFT algorithm takes the time signal and duplicates it from

minus to plus infinity. If the amplitude of the measured time signal differs



between the start and the end of the window, a discontinuity occurs during this

multiplication introducing an error in the FFT algorithm. This can be corrected

using a Hanning window. Holography used a double FFT so the AdHoc

window is used, which is basically a two dimensional Hanning window thus

removing discontinuities in the both the x and y directions.

The one-dimension Ad Hoc window (W) would be:

Section 2.6 Derivation of other acoustic quantities

If we know how the plane waves propagate, we can calculate the pressure field

in any parallel plane, by adding the contributions of all plane waves. This will

be correct only if all acoustic sources are on the same side of both planes :

Knowing the pressure field on the parallel plane, it is possible to calculate the

particle velocity and eventually the intensity on this plane.

The particle velocity (V) will be known if the pressure differential can be

determined -which is the case with Acoustic holography since the pressure can

be measured at r and (r + dr)


Rev 12A 15

Once the pressure and the velocity are known then the intensity is just the

product of the two.

Rev 12A 17

Index

(

(Back) propagating to other planes • 13

A

Acoustic holography concepts • 7

D

Derivation of other acoustic quantities • 15

I

Introduction • 5

S

Spatial domain • 8

T

Temporal and spatial frequency • 7

The Wiener filter and the AdHoc window • 14

Time domain • 7

Documents

Acoustic holography LMS Test