227-0477-00L

Le ture Notes on A ousti s I

Kurt Heuts hi

Institute for Signal- and Informationpro essing, ISI

Swiss Federal Institute of Te hnology, ETH

CH-8092 Zuri h

2020-08-07

The gure shows the interferen e pattern of two sound sour es lo ated at [-0.5,0.0 and [0.5,0.0 for a frequen y of 4.5 kHz. Bright

regions show high sound pressure. The lo al variation is highest on a line between the sour e points.

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Contents

1 A ousti fundamentals 1

1.1 Introdu tion: A ousti s and sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Basi sound wave phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Geometri al spreading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.2 Ree tion of sound waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.3 S attering of sound waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.4 Interferen e of sound waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.5 Dira tion of sound waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Fundamental quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3.1 Sound pressure, sound parti le displa ement and sound velo ity . . . . . . . . 4

1.3.2 Sound intensity and sound power . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3.3 Impedan e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.4 Volume velo ity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Fundamental equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4.1 Wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4.2 Sinusoidal waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4.3 Complex representation of sinusoidal quantities . . . . . . . . . . . . . . . . . 11

1.4.4 Helmholtz equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 Solutions of the wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5.1 Plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5.2 Spheri al waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.5.3 Cylindri al waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.6 Sound pressure and sound power for point sour es . . . . . . . . . . . . . . . . . . . . 15

1.7 Superposition of point sour es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.7.1 Superposition of in oherently radiating point sour es . . . . . . . . . . . . . . 16

1.7.2 Superposition of oherently radiating point sour es . . . . . . . . . . . . . . . 17

1.8 Ree tion of sound waves at a ousti ally hard surfa es . . . . . . . . . . . . . . . . . 18

1.8.1 Spe ular ree tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.8.2 Sour e dire tivity for limited radiation angles . . . . . . . . . . . . . . . . . . 19

1.8.3 Diuse ree tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.9 Doppler ee t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.10 Soni boom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.11 dB - s ale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.11.1 Quantities expressed as levels . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.11.2 Consequen es of the dB s ale . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.11.3 Subtlety of the dB s ale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.11.4 Computations involving dB quantities . . . . . . . . . . . . . . . . . . . . . . 22

1.11.5 Typi al values of sound pressure levels . . . . . . . . . . . . . . . . . . . . . . 22

1.12 Classi ation of a ousti al signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.13 Standing waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.13.1 Superposition of waves traveling in opposite dire tions . . . . . . . . . . . . . 24

1.13.2 Quarter wave length resonators . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.14 Sound eld al ulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.14.1 General problem of ree tion . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.14.2 Kir hho-Helmholtz Integral and Boundary Element Method . . . . . . . . . . 27

1.14.3 Appli ations of the Kir hho-Helmholtz Integral . . . . . . . . . . . . . . . . . 27

1.14.4 Method of Finite Dieren es . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

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1.14.5 Method of nite elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.14.6 A ousti al Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

1.14.7 Equivalent sour es te hnique . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

1.14.8 Prin iple of re ipro ity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

1.15 Produ tion of sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

1.15.1 Relaxation of ompressed air . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

1.15.2 Abrupt gas produ tion (explosion) . . . . . . . . . . . . . . . . . . . . . . . . 41

1.15.3 Modulated air ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

1.15.4 Os illating air olumn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

1.15.5 Vibrating bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

1.15.6 Thermo-a ousti al ma hines . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2 A ousti al measurements 48

2.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.2 Signal attributes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.2.2 Appli ation of the measurement attributes . . . . . . . . . . . . . . . . . . . . 50

2.2.3 Algorithm to determine the moving square average . . . . . . . . . . . . . . . 51

2.3 Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.3.1 Weighting lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.3.2 Filters for frequen y analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.4 Un ertainty of measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.4.1 Degrees of freedom of a bandlimited random signal . . . . . . . . . . . . . . . 54

2.4.2 Expe tation value and varian e of various fun tions of statisti ally independent samples 55

2.4.3 Un ertainty of the al ulation of the root mean square . . . . . . . . . . . . . 56

2.5 Measurement instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.5.1 Mi rophones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.5.2 Calibrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.5.3 Sound level meter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.5.4 Level re orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.5.5 Analyzers for level statisti s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.5.6 Frequen y analyzers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.5.7 Sound re orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.6 Spe ial measurement tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.6.1 Sound intensity measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.6.2 System identi ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.6.3 Measurement of reverberation times . . . . . . . . . . . . . . . . . . . . . . . 68

2.7 Pressure zone mi rophone onguration . . . . . . . . . . . . . . . . . . . . . . . . . 70

2.8 Un ertainty of a ousti al measurements . . . . . . . . . . . . . . . . . . . . . . . . . 72

3 The human hearing 73

3.1 Stru ture and prin iple of operation of the ear . . . . . . . . . . . . . . . . . . . . . . 73

3.2 Properties of the auditory system for stationary signals . . . . . . . . . . . . . . . . . 74

3.2.1 Loudness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.2.2 Frequen y dis rimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.2.3 Criti al bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.2.4 Audibility of level dieren es . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.2.5 Masking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.2.6 Loudness summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.2.7 Virtual pit h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.2.8 Audibility of phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.2.9 Methods to al ulate and measure the loudness . . . . . . . . . . . . . . . . . 78

3.2.10 Nonlinear distortions of the ear . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.3 Properties of the ear for non stationary signals . . . . . . . . . . . . . . . . . . . . . . 78

3.3.1 Loudness dependen y on the signal length . . . . . . . . . . . . . . . . . . . . 78

3.3.2 Temporal masking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.4 Binaural hearing: lo alization of sound sour es . . . . . . . . . . . . . . . . . . . . . . 79

3.4.1 Lo alization in the horizontal lane . . . . . . . . . . . . . . . . . . . . . . . . 80

3.4.2 Lo alization in the verti al plane (elevation) . . . . . . . . . . . . . . . . . . . 80

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3.4.3 Per eption of distan es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.4.4 E hoes and the pre eden e ee t . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.5 Hearing damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.5.1 Me hanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.5.2 Assessment of the danger for a possible hearing damage . . . . . . . . . . . . 82

4 Musi al Intervals 83

5 Outdoor sound propagation 84

5.1 Basi equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.2 Dire tivity of the sour e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.3 Attenuation terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.3.1 Geometri al divergen e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.3.2 Atmospheri absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.3.3 Ground ee t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.3.4 Obsta les . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.4 Ree tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.5 Meteorologi al ee ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.5.1 Temperature gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.5.2 Wind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.5.3 Favorable and unfavorable sound propagation onditions . . . . . . . . . . . . 94

5.5.4 Turbulen e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.5.5 Cal ulation of meteorologi al ee ts on sound propagation . . . . . . . . . . . 95

6 Absorption and ree tion 97

6.1 Chara terization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.2 Types of absorbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.2.1 Porous absorbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.2.2 Resonan e absorbers of type Helmholtz . . . . . . . . . . . . . . . . . . . . . 97

6.2.3 Membrane absorbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.3 Measurement of absorption and ree tion . . . . . . . . . . . . . . . . . . . . . . . . 100

6.3.1 Kundt's tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.3.2 Impedan e tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.3.3 Reverberation hamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.3.4 In situ measurement of impulse responses . . . . . . . . . . . . . . . . . . . . 102

6.4 Cal ulation of absorption and ree tion from impedan e relations . . . . . . . . . . . 102

6.4.1 Normal in iden e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.4.2 Oblique in iden e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.5 Typi al values of absorption oe ients . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.6 Cover for porous absorbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7 Room a ousti s 107

7.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.2 Room a ousti s of large rooms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.2.1 Statisti al room a ousti s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.2.2 Geometri al room a ousti s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.2.3 A ousti al design riteria for rooms . . . . . . . . . . . . . . . . . . . . . . . . 118

7.2.4 Room a ousti al design tools . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

7.2.5 Some room a ousti al ee ts that are not onsidered with statisti al or geometri al a ousti s122

7.2.6 Ree tions at spheri al surfa es . . . . . . . . . . . . . . . . . . . . . . . . . 124

7.3 Room a ousti s of small rooms, wave theoreti al a ousti s . . . . . . . . . . . . . . . 129

7.3.1 Wave equation and boundary onditions . . . . . . . . . . . . . . . . . . . . . 129

7.3.2 Solution for re tangular rooms with a ousti ally hard surfa es . . . . . . . . . 130

7.3.3 Sour e - re eiver transfer fun tion . . . . . . . . . . . . . . . . . . . . . . . . 133

7.3.4 A ousti al design of small rooms . . . . . . . . . . . . . . . . . . . . . . . . . 133

7.4 Room a ousti al measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

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8 Building a ousti s 139

8.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

8.2 Airborne sound insulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

8.2.1 Sound insulation index R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

8.2.2 Sound insulation of single walls . . . . . . . . . . . . . . . . . . . . . . . . . . 140

8.2.3 Sound insulation of double walls . . . . . . . . . . . . . . . . . . . . . . . . . 141

8.2.4 Standard sound pressure level dieren e . . . . . . . . . . . . . . . . . . . . . 141

8.3 Impa t sound insulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

8.4 SIA 181 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

8.5 Constru tion hints for good building a ousti al onditions . . . . . . . . . . . . . . . . 142

9 Noise abatement 143

9.1 Introdu tion - denition of noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

9.2 Ee ts of noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

9.3 General remarks for the assessment of noise . . . . . . . . . . . . . . . . . . . . . . . 144

9.4 Inuen e of the sour e type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

9.5 Denition of limiting values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

9.6 Legal basis in Switzerland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

9.6.1 Environment prote tion law USG . . . . . . . . . . . . . . . . . . . . . . . . . 145

9.6.2 Noise Abatement Ordinan e LSV . . . . . . . . . . . . . . . . . . . . . . . . . 145

9.7 Sounds ape on ept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

A A ousti physi al onstants 150

A.1 speed of sound in air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

A.2 density of air at sea level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

A.3 a ousti impedan e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

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Chapter 1

A ousti fundamentals

1.1 Introdu tion: A ousti s and sound

A ousti s is the s ien e of sound. Typi al questions deal with the generation of sound, the propagation

and intera tion with matter and the per eption by humans. The term sound stands for me hani al

os illations with wave-like propagation. Sound waves an propagate in air, in liquids or in solid bodies.

Figure 1.1 shows plane wave propagation in an open ended tube as a movie sequen e.

Figure 1.1: Movie pi tures of the sound propagation in a long and open tube. The sound waves are

generated by the moving piston shown on the left. The dots depi t air parti les. Of spe ial interest is the

lo al density of the parti les whi h orresponds to sound pressure and the speed of the parti les whi h

orresponds to sound parti le velo ity. It should be noted that on average there is no net movement of

the parti les.

Corresponding to the per eptional apabilities of the human ear, three dierent frequen y ranges are

distinguished. The range of hearing stret hes from about 16 Hz to 16 kHz. Lower frequen ies are

alled infra-sound, higher frequen ies are alled ultra-sound.

The eld of a ousti s an be subdivided into several spe ial topi s su h as:

Theoreti al a ousti s analyti al and numeri al methods for sound eld al ulations.

Nonlinear a ousti s nonlinear ee ts that o ur at events of extremely high sound pressure su h as

explosions or soni booms of obje ts that move faster than the speed of sound.

1

Underwater a ousti s sound propagation in water, sonar systems, seismi explorations.

Ultrasound non destru tive test pro edures for materials, medi al appli ations.

Vibrations vibrational behavior of bodies, sound radiation of vibrating stru tures.

Noise ontrol des ription and modeling of noise sour es, investigations on noise prote tion measures.

Room a ousti s assessment, planing and predi tion of sound elds in rooms.

Building a ousti s noise ontrol in buildings, transmission loss of building stru tures.

Ele troa ousti s transdu ers (mi rophones, loudspeakers), re ording devi es, publi address systems,

signal pro essing in a ousti s.

A ousti s of the ear stru ture of the ear, hara teristi s of the ear, per eption and subje tive evalu-

ation of noise.

1.2 Basi sound wave phenomena

1.2.1 Geometri al spreading

Sound wave fronts that origin from a sour e with nite extension spread with growing distan e over an

in reasing surfa e. Correspondingly the amplitude of the sound wave de reases (Figure 1.2).

Figure 1.2: Geometri al spreading of a pulse shaped sound wave (time progresses from left to right).

The lo ation dependent sound pressure is olor oded where intense red orresponds to high positive

values and intense blue stands for high negative values.

1.2.2 Ree tion of sound waves

If a sound wave hits an obje t, the free propagation is disturbed. At least a portion of the in ident

wave will be thrown ba k that i.e. will be ree ted. If the ree ting obje t is large and at, a spe ular

ree tion o urs. In this ase the billiard rule angle of in iden e = angle of ree tion holds. The

ree ted wave has a distin t orientation and has the same temporal hara teristi s as the in ident wave

(Figure 1.3).

Figure 1.3: Ree tion of a pulse shaped sound wave at a large smooth surfa e.

2

1.2.3 S attering of sound waves

If the size of the ree ting obje t is small or the ree ting surfa e is signi antly stru tured in depth

( ompared to the wave length), the ree tion is no longer spe ular but s attering or diuse. Ree ted

waves represent no distin t dire tion and are smeared over time (Figure 1.4).

Figure 1.4: Ree tion or s attering of a pulse shaped sound wave at several small obje ts.

1.2.4 Interferen e of sound waves

If two or more sound waves superpose, the resulting wave has sound pressure and sound velo ity orre-

sponding to the sum of the individual pressures and velo ities. This summation has to be understood

for ea h point in spa e and time. If the individual waves have identi al frequen y, their relative phase

de ides wether they amplify or attenuate ea h other. If the phase dieren e between two waves is small,

an ampli ation o urs and the interferen e is alled onstru tive. If the phase dieren e tends to 180

the waves attenuate ea h other and the interferen e is alled destru tive (Figure 1.5).

Figure 1.5: Superposition of two sound sour es emitting sinusoidal waves resulting in lo ation spe i

onstru tive or destru tive interferen e.

1.2.5 Dira tion of sound waves

Dira tion des ribes the phenomenon that waves are bent around obsta les (Figure 1.6). The dee tion

into the geometri al shadow is stronger for lower frequen ies.

Figure 1.6: Dira tion of a pulse shaped sound wave at an edge (time progresses from left to right).

The edge of the barrier is the origin of a se ondary wave.

3

1.3 Fundamental quantities

1.3.1 Sound pressure, sound parti le displa ement and sound velo ity

One of the onsequen es of the hull of air surrounding the earth is a stati pressure. This atmospheri

pressure is highest at sea level and de reases with height. On average the atmospheri pressure is

about 100'000 Pa. A variation in altitude of 1 m results in a hange of about 12 Pa. The atmospheri

pressure is superimposed by small u tuations as a onsequen e of sound waves. The human ear is only

sensitive to these variations. Consequently these fast u tuations relative to the atmospheri pressure

got a spe ial name. The orresponding quantity is alled sound pressure (dt: S halldru k) and is dened

as (Eq. 1.1):

p(t) = P (t)− Patm (1.1)

where

p(t): sound pressure

P (t): momentary air pressure

Patm: atmospheri pressure

The produ tion of lo al pressure variations leads to waves that travel with the speed of sound. Sound

waves transport energy by the intera tion of adja ent elements. Therefore they require matter with

a mass and spring hara teristi s. In air (airborne sound) sound waves are always longitudinal waves

whi h means that the gas parti les move ba k and forth in the propagation dire tion. The movement

of the gas parti les is des ribed by the sound parti le displa ement ζ (dt: S hallauss hlag) and by the

sound parti le velo ity ~v (dt: S halls hnelle). The sound parti le velo ity ~v is a ve tor whi h points in

the propagation dire tion. The displa ement and the velo ity are related by Eq. 1.2.

v(t) =dζ

dt(1.2)

Sound pressure and sound parti le velo ity represent the two fundamental quantities to des ribe

a ousti al pro esses.

A sound eld des ribes the a ousti al onditions in a region in spa e. A omplete des ription of a

sound eld requires in prin iple knowledge of sound pressure and sound parti le velo ity at every point

in spa e. However as sound parti le velo ity an be related to the sound pressure gradient (see below),

the velo ity eld an be al ulated from omplete information about sound pressure alone.

Typi al numeri al values

sound pressure normal spee h produ es in 1 m distan e typi al root mean squared sound pressure

values ptyp,rms of about 0.1 Pa. At frequen ies around 1 kHz sound pressure values pmin,rms of 2

×10−5Pa are just audible. The threshold of pain of the human auditory system is at pmax,rms ≈

100 Pa.

sound parti le displa ement at a frequen y of 1 kHz the above indi ated sound pressure values

orrespond to sound parti le displa ements of ζtyp,rms ≈ 4× 10−8m, ζmin,rms ≈ 8× 10−12

m and

ζmax,rms ≈ 4× 10−5m.

sound parti le velo ity in a plane wave the sound pressure values from above orrespond to the

following parti le velo ities: vtyp,rms ≈ 2.5 × 10−4m/s, vmin,rms ≈ 5 × 10−8

m/s and vmax,rms ≈0.25 m/s.

1.3.2 Sound intensity and sound power

The energy transport related to a sound wave an be des ribed by the sound intensity (dt: S hallinten-

sität)

~I. The intensity indi ates the amount of sound energy per unit time or sound power that passes

through an orthogonal unit area. The sound intensity is a ve tor and points in the same dire tion as

the sound parti le velo ity. The absolute value equals the produ t of sound pressure and sound parti le

velo ity (taking into a ount a possible phase shift).

I = pv [W/m2] (1.3)

4

The bar in Eq. 1.3 indi ates averaging in time. In the vi inity of sound sour es or ree tors there is

usually a phase shift between p and v. In extreme ases this an lead to low intensity values although

sound pressure and sound parti le velo ity have both high amplitudes. The physi al interpretation is

that air moves ba k and forth without signi ant ompression. In other words there is a lot of rea tive

power but only little ee tive power.

If the sound intensity is known, the sound power W passing through an area S is given by the integral

in Eq. 1.4.

W =

∫

S

~Id~S [W ] (1.4)

The multipli ation denotes the s alar produ t of the intensity ve tor

~I and the orthogonal ve tor of

the area element d~S. The sound power W orresponds to the total radiated power of the sour e if the

area S en loses the sour e ompletely.

The sound power of typi al sour es is very small as shown in Table 1.1.

sound power [W

human voi e, normal 7×10−6

human voi e, max. 2×10−3

violin, fortissimo 1×10−3

Hi-Fi loudspeaker (10 W el.) 0.1

ja khammer 1

organ, fortissimo 10

or hestra (75 persons) 70

airplane Boeing 747 6'000

airplane FA-18 200'000

Table 1.1: Examples of sound sour es and their emitted sound power.

1.3.3 Impedan e

The ratio of sound pressure and sound parti le velo ity is dened as a ousti al impedan e Z (dt.

Impedanz).

Z =p

v(1.5)

The symbol . stands for the omplex amplitude that ontains an amplitude and a phase information.

In general the impedan e Z is a omplex quantity.

1.3.4 Volume velo ity

In the dis ussion of sound radiation the quantity volume velo ity Q (dt: S halluss) plays an important

role. It indi ates the amount of sound that passes through a ertain area (Eq. 1.6). The multipli ation

stands for the s alar produ t of the sound parti le velo ity and orthogonal ve tor of the area element

d~S.

Q =

∫

S

~vd~S (1.6)

1.4 Fundamental equations

1.4.1 Wave equation

The wave equation is the fundamental dierential equation that des ribes in a ompa t form the physi s

of sound elds. For their derivation the intera tions between sound pressure and sound parti le velo ity

will be formulated.

5

Intera tion between sound pressure and sound parti le velo ity

The ee t of sound pressure on sound parti le velo ity is investigated in a small ube with dimensions

∆l ·∆l ·∆l. It is assumed that the sound pressure p on all six fa es of the ube is known. Given this

we are looking for the behavior of the sound parti le velo ity ~v in the ube (Fig. 1.7).

Figure 1.7: Situation to investigate the intera tion between sound pressure p and sound parti le velo ity~v in a small ube.

The onsequen e of pressure dieren es on opposite sides of the ube is an a eleration a of the air

with mass m in between. On e the a eleration is known, the sound parti le velo ity an be dedu ed

easily. The relevant physi al equation is Newton's law (1.7).

Fres = m · a (1.7)

The resulting for e Fres orresponds to the pressure dieren e multiplied by the area. The a eleration

equals the time derivative of the sound parti le velo ity in the orresponding dire tion. Here this is

shown for the x oordinate dire tion (1.8).

∆l2(px0 − px1) = m∆vx∆t

(1.8)

The mass of the ube is related to density ρ as:

m = ∆l3 · ρ (1.9)

Eq. 1.8 be omes

∆l2(px0 − px1) = ∆l3 · ρ∆vx∆t

(1.10)

Finally with division by the volume of the ube ∆l3 it follows from Eq. 1.10

px0 − px1∆l

= ρ∆vx∆t

(1.11)

Eq. 1.11 an be written as separated dierential equations for the three dire tions in spa e:

∂p

∂x= −ρ∂vx

∂t∂p

∂y= −ρ∂vy

∂t

∂p

∂z= −ρ∂vz

∂t(1.12)

or in ve tor equation form:

grad(p) = −ρ∂~v∂t

(1.13)

6

Intera tion between sound parti le velo ity and sound pressure

The ee t of sound parti le velo ity on sound pressure is again investigated in a small ube with

dimensions ∆l ·∆l ·∆l. It is assumed that the sound parti le velo ity ~v is given on all six fa es of the

ube. We are looking for the behavior of the sound pressure p in the ube (Fig. 1.8).

Figure 1.8: Situation to investigate the intera tion between sound parti le velo ity ~v and sound pressurep in a small ube.

A dieren e in sound parti le velo ity on two opposite sides of the ube results in a hange of the ube

volume ∆V . This volume hange is onne ted to a hange in pressure ∆P . Assuming an adiabati

pro ess, the relation between ∆V and ∆P is des ribed by the Poisson law (1.14). The assumption of

an adiabati pro ess is usually fullled for sound in air. This implies that there is no heat ex hange

between the sound wave and the surrounding. However in spe ial ases su h as a loudspeaker box

lled with porous material the pro ess is no longer adiabati but isothermal. For adiabati pro esses

an expansion of the gas leads to a pressure de rease and a ooling of the gas.

PV κ = constant (1.14)

with

P : pressure of the gasV : volume

κ: adiabati exponent, for air κ = 1.4

For small variations the Poisson law in Eq. 1.14 an be linearized. A small pressure variation ∆P is

related to a small volume hange ∆V :

(P0 +∆P )(V0 +∆V )κ = P0Vκ0 (1.15)

The rst term in Eq. 1.15 an be rewritten as

P0 +∆P = P0

(

1 +∆P

P0

)

(1.16)

For small hanges ∆V ompared to V the expression (V0 + ∆V )κ an be expanded into a series.

Ignoring the higher order elements of the series one gets:

(V0 +∆V )κ ≈ V κ0 +∆V κV κ−10 = V κ0

(

1 + κ∆V

V0

)

(1.17)

(1.15) and (1.17) in (1.14) yields:

P0

(

1 +∆P

P

)

V κ0

(

1 + κ∆V

V0

)

≈ P0Vκ0 (1.18)

(

1 +∆P

P0

)(

1 + κ∆V

V0

)

≈ 1 (1.19)

∆P

P0≈ −κ∆V

V0− κ

∆P

P0

∆V

V0(1.20)

7

The produ t ∆P∆V an be ignored under the assumption of small quantities. So nally we get:

∆P

P0≈ −κ∆V

V0(1.21)

The linearized form of the Poisson Equation (1.21) onne ts in a simple way the pressure variation and

the volume variation due to the sound parti le velo ity dieren es on all sides of the ube. Let the

volume of the ube at time t be

V (t) = V0 = ∆l3 (1.22)

Short time later the volume is

V (t+∆t) = [∆l +∆t(vx1 − vx0)] · [∆l +∆t(vy1 − vy0)] · [∆l +∆t(vz1 − vz0)] (1.23)

The produ ts of two and three sound parti le velo ity dieren es be ome very small and an be ne-

gle ted:

V (t+∆t) ≈ ∆l3 +∆l2∆t(vx1 − vx0) + ∆l2∆t(vy1 − vy0) + ∆l2∆t(vz1 − vz0) (1.24)

The volume hange ∆V during the time step ∆t is

∆V = V (t+∆t)− V (t) ≈ ∆l2∆t(vx1 − vx0) + ∆l2∆t(vy1 − vy0) + ∆l2∆t(vz1 − vz0) (1.25)

Insertion of Eq. 1.25 in Eq. 1.21 gives

∆P =−κP0

∆l3[

∆l2∆t(vx1 − vx0) + ∆l2∆t(vy1 − vy0) + ∆l2∆t(vz1 − vz0)]

(1.26)

or

∆P

∆t= −κP0

(

vx1 − vx0∆l

+vy1 − vy0

∆l+vz1 − vz0

∆l

)

(1.27)

It should be noti ed that the variation of the pressure ∆P equals the sound pressure hange ∆p.Translated into a dierential equation, Eq. 1.27 results in

∂p

∂t= −κP0

(

∂vx∂x

+∂vy∂y

+∂vz∂z

)

(1.28)

or abbreviated

∂p

∂t= −κP0div(~v) (1.29)

The two equations 1.12 and 1.28 represent the fundamental physi al relations for a ousti al pro esses.

The wave equation ombines these two relations into one single dierential equation. For its derivation,

the equations 1.12 are dierentiated with respe t to the three oordinates x, y and z, Eq. 1.28 is

dierentiated regarding to t. One gets:

∂2p

∂x2= −ρ ∂

2vx∂t∂x

=∗) −ρ ∂2vx∂x∂t

∂2p

∂y2= −ρ∂

2vy∂t∂y

=∗) −ρ∂2vy∂y∂t

∂2p

∂z2= −ρ ∂

2vz∂t∂z

=∗) −ρ ∂2vz∂z∂t

(1.30)

*) theorem of S hwarz

and

∂2p

∂t2= −κP0

(

∂2vx∂x∂t

+∂2vy∂y∂t

+∂2vz∂z∂t

)

(1.31)

insertion of 1.30 in 1.31 results in the wave equation:

8

∂2p

∂t2=κP0

ρ

(

∂2p

∂x2+∂2p

∂y2+∂2p

∂z2

)

(1.32)

or

∂2p

∂x2+∂2p

∂y2+∂2p

∂z2=

ρ

κP0

∂2p

∂t2(1.33)

The wave equation (1.33) represents a relation between the derivatives of sound pressure with respe t

to spa e and time. From experien e follows that a lo al sound eld disturban e propagates as a sound

wave. It is postulated that the disturban e propagates with the speed of sound c. The one-dimensional

sound eld an be written as an arbitrary fun tion with argument of form (x− ct) where x is the spa e

oordinate and t is time. Insertion into the wave equation (1.33) yields for the speed of sound c:

1 =ρ

κP0c2 (1.34)

or

c =

√

κP0

ρ(1.35)

It turns out that c is almost independent of pressure and density as these two quantities ompensate

ea h other to large extent in the term P0/ρ. The speed of sound is almost identi al on top of the

Himalaya and at sea level. The impedan e on the other hand is onsiderably lower at high altitudes

whi h means that the sound pressure produ ed by a vibrating body is smaller.

With the speed of sound c the wave equation for the sound pressure p an be written as

p = 1

c2∂2p

∂t2(1.36)

where

p: three dimensional Lapla e operator.

For a artesian oordinate system the Lapla e operator is

p = ∂2p

∂x2+∂2p

∂y2+∂2p

∂z2(1.37)

For ylindri al oordinates the Lapla e operator is given by

p = ∂2p

∂r2+

1

r

∂p

∂r+

1

r2∂2p

∂φ2+∂2p

∂z2(1.38)

The wave equation is the basis for the des ription of sound elds. The equation has to be fullled for

ea h point in spa e. The spe i ation of a on rete problem makes it ne essary to indi ate boundary

onditions su h as the velo ity distribution of a vibrating surfa e or the a ousti al impedan e of eld

limiting areas. The solution for the sound eld is found as the fun tion that fullls both the wave

equation and the boundary onditions at the same time.

For the appli ation of the wave equation one has to bear in mind that the equations used for its

derivation were found by linearization of the fundamental physi al equations. Stri tly speaking the

wave equation is no longer valid for high pressure or velo ity values. An expli it appli ation of the non

linear behavior of air is the usage of modulated high frequen y sound for publi address systems. The

high frequen y waves that an be emitted fo used to a narrow angle in spa e demodulate in the air

and produ e in this way the hearable audio signal

1

.

The speed of sound c and the density ρ0 of the air depend on temperature. As good approximation

one an write

1

F. Joseph Pompei, The Use of Airborne Ultrasoni s for Generating Audible Sound Beams, Journal of the Audio

Engineering So iety, vol. 47, p. 726-731 (1999).

9

c ≈ 343.2

√

T

293(1.39)

and

ρ0 ≈ ρrefPaT0P0T

(1.40)

where

T : temperature in Kelvin

Pa: air pressure in Pas al [Pa

T0: 293 K

P0: 101325 Pa

ρref : 1.186 kg/m

3

A more a urate expression (1.42) for the speed of sound an be found by taking into a ount the

parameters temperature, pressure, humidity and CO

2

on entration

2

. Besides the temperature inuen e

there is a weak dependen y on humidity (Fig. 1.9).

c(t, P, xw , xc) = a0 + a1t+ a2t2 + (a3 + a4t+ a5t

2)xw + (a6 + a7t+ a8t2)P (1.41)

+(a9 + a10t+ a11t2)xc + a12x

2w + a13P

2 + a14x2c + a15xwPxc

where

t: temperature in degrees Celsius

P : air pressure in Pas al

xw: water vapour mole fra tion, where xw ≈ (h/P )(1.00062 + 3.14 × 10−8P + 5.6 ×10−7t2)exp(1.2811805× 10−5T 2 − 1.9509874× 10−2T + 34.04926034− 6.3536311× 103T−1)h: relative humidity as a fra tion (0 < h < 1)T : temperature in Kelvin = t+ 273.15xc: CO

2

mole fra tion, typi al value: = 0.000314

a0 = 331.5024, a1 = 0.603055, a2 = −0.000528, a3 = 51.471935, a4 = 0.1495874, a5 =−0.000782, a6 = −1.82 × 10−7, a7 = 3.73 × 10−8, a8 = −2.93 × 10−10, a9 = −85.20931, a10 =−0.228525, a11 = 5.91 × 10−5, a12 = −2.835149, a13 = −2.15 × 10−13, a14 = 29.179762, a15 =0.000486

The formula is valid for t between 0 and 30

C, for P between 75'000 and 102'000 Pa and for xw

between 0 and 0.06.

1.4.2 Sinusoidal waves

Waves with sinusoidal time dependen y play an important role for theoreti al onsiderations. Su h

waves are hara terized by their frequen y f or their angular frequen y ω or their period length T .

f =1

T(1.42)

ω = 2πf (1.43)

A xed point on a sinusoidal wave train travels one wave length λ within the time T (Fig. 1.10).

Therefore

λ = cT =c

f(1.44)

Often usage of wave number k it is helpful where

k =2π

λ(1.45)

2

Owen Cramer, The variation of the spe i heat ratio and the speed of sound in air with temperature, pressure,

humidity, and CO

2

on entration, Journal of the A ousti al So iety of Ameri a, vol. 93, p.2510-2516 (1993).

10

0 20 40 60 80 100

rel. humidity [%]

330

335

340

345

350

355

so

un

d s

peed

[m

/s]

t = 0°

t = 10°

t = 20°

t = 30°

Figure 1.9: Speed of sound c for an air pressure of 1013 hPa as a fun tion of air humidity with

temperature as parameter.

Figure 1.10: Sinusoidal wave with period length T in time and wave length λ in spa e.

1.4.3 Complex representation of sinusoidal quantities

Quantities with sinusoidal behavior may be represented as pointers in the omplex plane. The pointer

has a ertain length - orresponding to the amplitude - and rotates a ording to angular frequen y with

onstant angular velo ity ounter lo kwise. The angle of the pointer at t = 0 orresponds to the initial

phase φ. The pointer marks a omplex number with an imaginary part that des ribes the sine fun tion

of the quantity (Fig. 1.11). The real part des ribes the orresponding osine fun tion.

The quantity p with sinusoidal variation:

p(t) = p sin (ωt+ φ) (1.46)

is represented by the pointer p:

p(t) = pej(ωt+φ) (1.47)

Cal ulations with omplex pointers are often easier to perform than dealing with sine and osine fun -

tions.

1.4.4 Helmholtz equation

With the restri tion to sinusoidal time dependen ies, the wave equation simplies to the Helmholtz

equation. The sinusoidal ex itation of a sound eld (assumed to be linear) yields sinusoidal time

dependen ies for all eld variables. It is therefore su ient to indi ate the amplitudes and phase

relations in ea h eld point.

11

Im

Re

Figure 1.11: Representation of a quantity with sinusoidal dependen y as imaginary part of a rotating

pointer in the omplex plane.

In omplex writing the sound pressure p an be written as produ t of a omplex, lo ation-dependent

amplitude fun tion p(location) and an os illation term ejωt (Eq. 1.48).

p(location, t) = p(location)ejωt (1.48)

For the Lapla e operator an be written

p = pejωt (1.49)

and

∂2p

∂t2= −ω2p(location)ejωt (1.50)

Insertion of (1.49) and (1.50) in (1.36 ) yields the Helmholtz equation (1.51):

p+ ω2

c2p = 0 (1.51)

The omplex amplitude fun tion p is only a fun tion of the position in spa e.

1.5 Solutions of the wave equation

1.5.1 Plane waves

A plane wave is the simplest wave type. The sound eld variables p and ~v are both in phase and

depend only on one spa e oordinate. For propagation in the x-dire tion, all points in the y, z plane

have identi al values of p and ~v. Most relevant for plane waves is the fa t that there is no geometri al

divergen e. Plane waves o ur e.g. in tubes with a diameter that is mu h smaller than the wave

length. Far away from sour es of limited size, the waves an usually be approximated as plane waves

with good a ura y.

The solutions of the one-dimensional wave equation

∂2p

∂x2=

1

c2∂2p

∂t2(1.52)

represent the set of possible sound pressure dependen ies. All fun tions p(x, t) that fulll Eq. 1.52

have the form

p(x, t) = f(ct± x) (1.53)

In the above equation f stands for an arbitrary fun tion. The one dimensional wave equation is thus

fullled if the argument of f has the form ct± x. A ertain value of the argument an be obtained by

12

adapting the time or the spa e variable - time and spa e are thus ex hangeable. The minus sign in the

argument stands for a wave propagating in positive x dire tion (to the right), a plus sign is a wave in

the negative x dire tion (to the left).

It is often onvenient to des ribe the arbitrary fun tion f as the superposition of sine waves a ording to

the theorem of Fourier. It is usually su ient to solve a ertain problem by investigating the behavior

for a sine wave of arbitrary frequen y. In omplex representation a ording to 1.48 we an write for the

sound pressure p:

p(x, t) = pej(−kx+φ)ejωt (1.54)

where

p: amplitude of the sine os illation

φ: onstant phase term

The sound parti le velo ity an be determined from sound pressure with Eq. (1.12). The plane wave in

the x dire tion auses the air parti les to move ba k and forth in the x dire tion. There is no movement

in the y and the z dire tion. The sound parti le velo ity in x is in omplex notation

vx(x, t) = vxejωt

(1.55)

where

vx: omplex, lo ation dependent amplitude fun tion.

Inserting (1.54) and (1.55) in (1.12) yields

pjkej(−kx+φ)ejωt = vxρjωejωt

(1.56)

With ω = kc one gets

vxejωt =

1

ρcpej(−kx+φ)ejωt (1.57)

or

vx(x, t) =1

ρcp(x, t) (1.58)

In a plane wave sound pressure and sound parti le velo ity are in phase and the ratio of their amplitudes

( orresponding to the impedan e Z0) is

Z0 = ρc (1.59)

1.5.2 Spheri al waves

Spheri al waves an be thought of emitted by a point sour e. They propagate spheri ally in all

dire tions. The two dimensional analogue are water waves that o ur as the results of a lo al

distortion, for example a stone falling into the water. Due to symmetry reasons the sound pressure and

the amplitude of the sound parti le velo ity have to be onstant on a spheri al surfa e with arbitrary

radius and a enter that oin ides with the sour e point. The ve tor of the sound parti le velo ity

points in radial dire tion outwards.

As a guess for the solution of the sound pressure in spheri al waves the approa h for plane waves is

assumed and omplemented with a 1/r (r: Radius) amplitude dependen y (1.60).

p(r, t) =1

rpej(−kr+φ)ejωt (1.60)

The validity of Eq. 1.60 an be proved with help of the Helmholtz equation. In spheri al oordinates

the equation for a sound eld variable that depends on the radius only is

∂2p

∂r2+

2

r

∂p

∂r+ k2p = 0 (1.61)

13

In analogy to plane waves the sound parti le velo ity an be dedu ed from sound pressure with help of

Eq. 1.12. The radial omponent is found as

vr(r, t) = p(r, t)

(

1

ρc+

1

jωρr

)

(1.62)

For the impedan e ZK of spheri al waves follows

ZK = ρcjkr

1 + jkr(1.63)

ZK depends on frequen y and distan e. For large distan es ( ompared to the wave number k) ZKapproa hes the value of plane waves. If the distan e gets small and smaller both real and imaginary

part of ZK drop o (Fig. 1.12).

0.0001

0.001

0.01

0.1

1

0.01 0.1 1 10 100 1000

kr

Z/Z

o

R e

Im

Figure 1.12: Real and imaginary part of the impedan e for spheri al waves. The abs issa is s aled as

produ t kr with k: wave number = 2π/λ and r: distan e. The ordinate shows the impedan e relative

to the value for plane waves Z0 = ρc.

The model for an ideal sour e that emits spheri al waves is a small (relative to the wave length) pulsating

sphere. A sphere with radius r0 and surfa e velo ity vr in radial dire tion produ es a volume velo ity

Q of

Q = 4πr20vr (1.64)

With Eq. 1.63 and under the assumption kr0 ≪ 1, the sound pressure on the surfa e of the sphere is

found as

p(r0) = vr(r0)ρcjkr0 (1.65)

Following Eq. 1.60 the spheri al wave approa h an be rewritten as follows with the stipulation that

the phase is now referred to the surfa e of the sphere:

p(r, t) =1

rpej(−k(r−r0)+φ)ejωt (1.66)

Comparison of Eq. 1.65 with 1.66 yields

1

r0pej(φ) = vr(r0)ρcjkr0 (1.67)

or

pej(φ) =Q

4πρcjk (1.68)

Amplitude and phase an be found as:

14

p =Q

4πρck

φ =π

2(1.69)

Considering the fa t that r0 is very small ompared to the wave length, the dieren e r − r0 in Eq.

1.66 an be approximated as r. With this the sound pressure p(r) at distan e r from a point sour e

with volume velo ity Q is found as

p(r) =jkρcQe−jkr

4πr(1.70)

1.5.3 Cylindri al waves

Sound pressure, sound parti le velo ity and impedan e for ylindri al waves an be determined analo-

gously to the ase of spheri al waves. The sound pressure dependen y with distan e r results as

pcyl. ∼1√r

(1.71)

Similarly to spheri al waves, ylindri al waves show a near eld and a far eld. However, the transition

is at kr ≈ 1 in ontrast to kr ≈ 2 for spheri al waves.

1.6 Sound pressure and sound power for point sour es

By denition, ideal point sour es radiate sound equally in all dire tions. If the sound pressure p(r) atdistan e r in the far eld (not too lose to the sour e) is known, the sound power W of the sour e an

be found as follows:

The impedan e Z in the far eld is

Z =p(r)

v(r)= ρc (1.72)

The intensity Irms(r) at distan e r from the sour e is

Irms(r) = prms(r)vrms(r) =p2rms(r)

ρc(1.73)

The totally emitted sound power an be found by integration of the intensity (1.73) over a losed

surfa e S that en loses the sour e.

W =

∫

S

~Id~S (1.74)

Most naturally, S is hosen as surfa e of a sphere with enter at the position of the sour e. In this ase

the intensity is onstant over S and the integration yields:

W = Irms(r)4πr2 =

p2rms(r)

ρc4πr2 (1.75)

1.7 Superposition of point sour es

Here the sound pressure at a re eiver position is investigated in the ase of several a tive sound sour es.

As long as the amplitudes of the sound eld variables are not too large (linear ase) the superposition

prin iple holds for sound pressure and sound parti le velo ity. This implies that the sound pressure at

a re eiver orresponds to the sum of the sound pressures resulting for ea h single sour e.

Two ases have to be distinguished:

15

In the rst ase the sour es radiate oherently, that is to say there is a xed phase relation between the

sour es. Here the resulting pressure equals the phase sensitive addition of the pressure ontributions of

ea h sour e. It is most bene ial to perform this addition using the omplex representation of sound

pressure.

In the se ond ase the sour es radiate in oherent signals, that is to say it is impossible to on lude

from the time signal of one sour e to the time signal of any other sour e. In this ase the superposition

simplies in the sense that intensities an be summed up. The resulting mean squared pressure equals

the sum of the squared pressure ontributions of ea h sour e.

1.7.1 Superposition of in oherently radiating point sour es

In oherently radiating point sour es distributed along a straight line

An innite row of equally distributed point sour es along a straight line is onsidered (Fig. 1.13).

a

dQ0 Q1Q-2

E

Figure 1.13: Situation of an innite row of in oherently radiating point sour es Q−∞ . . .Q+∞. The

re eiver E is lo ated in distan e a from the line.

The mean squared pressure p2rms,n at the re eiver E aused by sour e n is

p2rms,n =K

a2 + (nd)2(1.76)

where

K: onstant to des ribe the sour e strength

The superposition of all sour es yields

p2rms,tot =

+∞∑

n=−∞

p2rms,n = K

+∞∑

n=−∞

1

a2 + (nd)2= K

1

d2

+∞∑

n=−∞

1a2

d2 + n2(1.77)

From symmetry follows that the sum from −∞ to +∞ in Eq. 1.77 an be written as two times the

sum from 1 to +∞ and a orre tion for the term for n = 0.

A good formulary tells us that

cothx =1

x+

2x

π2

+∞∑

n=1

1x2

π2 + n2(1.78)

With substitution of

xπ = a

d , Eq. 1.78 an be rewritten as

+∞∑

n=1

1a2

d2 + n2=πd

2acoth

(πa

d

)

− d2

2a2(1.79)

Finally Eq. 1.77 an be written as

p2rms,tot =K

d2πd

acoth

(πa

d

)

=Kπ

adcoth

(πa

d

)

(1.80)

For the dis ussion of Eq. 1.80, two ases have to be distinguished:

16

πad small (small distan es) in this ase the approximation holds: coth

(

πad

)

≈ dπa . It follows for

p2rms,tot ≈ Ka2 or prms,tot ≈

√Ka . The dependen y of p(a) with distan e orresponds to 1/a, just

as for a point sour e. In the proximity of the row of point sour es the pressure at the re eiver is

dominated just by the sour e that is nearest.

πad large (large distan es) in this ase the approximation holds: coth

(

πad

)

≈ 1. It follows for

p2rms,tot ≈ Kπad or prms,tot ≈

√

Kπd

1√a. The dependen y of p(a) with distan e orresponds to

1/√a, just as for a line sour e.

The transition between the two distan e regimes an be lo alized where the two approximations yield

the same result:

a =d

π(1.81)

In oherent radiating point sour es distributed along a line of nite length

If the row of point sour es has limited length, there is for large distan es a transition from the line

sour e behavior to a point sour e behavior. The mathemati al proof is easiest if the separation between

the point sour es tends to 0. The summation orresponds then to an integration over a distin t range.

The nal result is a transition distan e a = L/π if L is the length of the point sour e row.

In oherent radiating point sour es distributed over an area of nite size

The distan e dependen y of an in oherent radiating re tangular area of length L and width B (L > B) an be des ribed by three regions a ording to Table 1.2.

a < B/π plane wave behavior (sound pressure independent of distan e)

B/π < a < L/π line sour e behavior

L/π < a point sour e behavior

Table 1.2: Distan e dependen y of sound pressure for in oherent radiating re tangular areas. a depi ts

the distan e, L is the length and B is the width of the radiating area.

1.7.2 Superposition of oherently radiating point sour es

Dipole radiator

The dipole radiator onsists of two oherently radiating point sour es of equal amplitudes but opposite

phase. The sound pressure at a re eiver point is given as the phase sensitive addition of the ontributions

of the two point sour es (Fig. 1.14).

With (1.60) the sound pressure in E an be written as:

p(r, t) = p

(

1

r1e−jkr1 − 1

r2e−jkr2

)

ejωt (1.82)

At low frequen ies and in the far eld, that is to say for ∆r ≪ r and k∆r ≪ 1, r1 and r2 an be

approximated as:

r1 ≈ r − ∆r

2(1.83)

r2 ≈ r +∆r

2(1.84)

With this follows

17

r1

r

r2+

-

E

φ∆ro

∆r

Figure 1.14: Geometry of a dipole with the re eiver E.

p(r, t) ≈ pe−jkr

r

(

ejk∆r/2

1− ∆r2r

− e−jk∆r/2

1 + ∆r2r

)

ejωt (1.85)

= pe−jkr

r

(

cos(k∆r/2) + j sin(k∆r/2)

1− ∆r2r

− cos(−k∆r/2) + j sin(−k∆r/2)1 + ∆r

2r

)

ejωt

Making use of approximations for small arguments (cos ǫ ≈ 1 and sin ǫ ≈ ǫ) yields

p(r, t) ≈ pe−jkr

r

(

1 + jk∆r/2

1− ∆r2r

− 1− jk∆r/2

1 + ∆r2r

)

ejωt (1.86)

= pe−jkr

r

(

(

1 + jk∆r2

) (

1 + ∆r2

)

−(

1− jk∆r2

) (

1− ∆r2

)

1−(

∆r2r

)2

)

ejωt

Under the far eld assumption, the denominator in the bra kets on the right hand side an be approxi-

mated as 1:

p(r, t) ≈ pe−jkr

r

(

∆r

r(r + jkr)

)

ejωt (1.87)

With ∆r ≈ ∆r0 cosφ follows:

p(r, t) ≈ p∆r0 cosφ

r2(r + jkr)e−jkrejωt (1.88)

For k ≫ 1 that is to say f ≫ 50 Hz Eq. 1.88 simplies to

p(r, t) ≈ pjk∆r0 cosφ

re−jkrejωt (1.89)

It should be noted that the amplitude term in (1.89) is proportional to k and therefore to frequen y.

The dipole radiation is very ine ient at low frequen ies.

1.8 Ree tion of sound waves at a ousti ally hard surfa es

1.8.1 Spe ular ree tion

The presen e of an a ousti ally hard surfa e implements a boundary ondition for the normal omponent

of the sound parti le velo ity with

vn = 0 (1.90)

18

An elegant on ept to deal with su h a boundary ondition is the introdu tion of one or more additional

equivalent sour es. These sour es are set and adjusted in order for their superposition with the original

sour e to satisfy the boundary ondition. With this in mind the ree tion of sound waves at a large,

a ousti ally hard surfa e an be treated with the on ept of a mirror sour e. The orresponding

additional sour e is pla ed at the mirrored position of the original sour e. The ee t of the ree tor

an then be repla ed by the ontribution of this additional sour e. The mirror sour e emits the same

signal as the original sour e (Fig. 1.15).

d=

2d

Figure 1.15: Repla ement of a ree ting surfa e by a mirror sour e.

1.8.2 Sour e dire tivity for limited radiation angles

Sour es lo ated lose to a ousti ally hard surfa es no longer radiate in all dire tions. For a broadband

sour e the limitation of radiation to a solid angle Φ results in an ampli ation orresponding to the

ratio 4π/Φ. For example, a sour e next to a orner appears with an sound power amplied by a fa tor

8.

1.8.3 Diuse ree tion

The dire tivity of diuse ree tions of sound waves is often des ribed by Lambert's law, originally

developed in Opti s. It assumes that the ree tion intensity I(φ) in dire tion φ is independent of the

in ident dire tion and proportional to the osine of φ (1.91) where φ is understood relative to the

normal dire tion (Fig. 1.16).

I(φ) = I0 cosφ (1.91)

φIo

Figure 1.16: cosφ dependen y of the intensity of a diuse ree tion assuming Lambert's law.

1.9 Doppler ee t

In ase of moving sour es or moving re eivers (relative to ea h other) a frequen y shift o urs. This

ee t is named after Ch. Doppler (1803-1852, Vienna) who dis overed and explained the phenomenon.

The ee t is omnipresent in daily life, for example in onne tion with passing ars. The Doppler ee t

plays an important role in sound radiation by loudspeakers that onsists of only one membrane.

The mathemati al dis ussion shall be based on the situation in Fig. 1.17. A point sour e Q is in x = 0at time t = 0. Q moves in positive x dire tion with speed vQ. It is assumed that Q emits a pure tone

19

of frequen y f0. We are looking for the frequen y f that is registered at a re eiver point E at distan e

d under an angle φ relative to the x dire tion.

Figure 1.17: Situation to investigate the Doppler frequen y shift.

The frequen y f is determined by evaluation of the time interval T between two sound pressure maxima.

A sound pressure maximum emitted at position Q rea hes the re eiver at time t = d/c. The next soundpressure maximum is emitted at positionQ′

at time t = T0 = 1/f0. Consequently this maximum rea hes

the re eiver at time t = T0 + d′/c. With this the time interval between two maxima at the re eiver is

T = T0 +d′

c− d

c(1.92)

For the frequen y f at the re eiver position follows:

f =1

1f0

− d−d′c

(1.93)

where d′ is found as

d′ =√

d2 − 2dvQT0 cosφ+ v2QT20 (1.94)

If the re eiver is lo ated on the x-axis (φ = 0), Eq. 1.94 simplies to

d′ = d− vQT0 (1.95)

and Eq. 1.93 be omes

f = f0c

c− vQ(1.96)

1.10 Soni boom

Sour es that move faster than the speed of sound produ e a soni boom. Typi al examples are air

planes, proje tiles or the end of a whip ord in a tion. Fig. 1.18 shows the development of su h a boom

(Ma h's one). At time 0 the sour e is in position Q0. After time t the sour e has rea hed position

Q3. After time t the wave front emitted in Q0 orresponds to a sphere of radius ct. The wave fronts

emitted from sour e positions between Q0 and Q3 are orrespondingly smaller spheres. The envelope

of all wave fronts forms a one with very high sound pressure. The opening angle α of the one is

sinα =c

v(1.97)

The tip of the one moves with the sour e. At the moment where the one rea hes the re eiver, a

sharp bang is heard.

20

Q 3

vt

ct

Q 0 Q 1

α

Figure 1.18: Development of Ma h's one for a sour e moving faster than the speed of sound.

1.11 dB - s ale

1.11.1 Quantities expressed as levels

An important hara terization of the behavior of a system is the ratio of the power at the output yand the power at the input x. Instead if indi ating this ratio linearly, often the logarithm of base 10 is

used. The orresponding unit is [Bel.

log10

(

powerYpowerX

)

[Bel (1.98)

The Bel s ale is very oarse. It is often more appropriate to introdu e a fa tor of 10 yielding tenth of

a Bel or de ibel [dB.

10 log10

(

powerYpowerX

)

[dB (1.99)

It is very ommon to express the a ousti quantities su h as sound pressure, sound intensity and

sound power in the de ibel s ale. If doing so the quantities get the name level as an appendix (sound

pressure level, ...). One of the reasons to use the dB s ale is the fa t that the sensation of the

human ear follows basi ally a logarithmi law. To express sound eld variables as levels, they have

to be onverted to power proportional quantities (if ne essary) and related to referen e values as follows:

Sound pressure level Lp

p0 = 2 × 10−5Pa is hosen as sound pressure referen e value. This orresponds to the threshold of

hearing at 1 kHz.

Lp = 10 log10

(

(

prms

p0

)2)

[dB (1.100)

Sound intensity level LI

The referen e value for sound intensity is I0 = 10−12W/m2.

LI = 10 log10

(

I

I0

)

[dB (1.101)

Sound power level LW

The referen e value for sound power is W0 = 10−12W .

LW = 10 log10

(

W

W0

)

[dB (1.102)

21

The above referen e values are hosen in su h a way that for a plane wave the sound pressure level

and the sound intensity level mat h within 0.1 dB. For a point sour e follows from (1.75), (1.100) and

(1.102) that sound power level and sound pressure level are identi al in a distan e of approximately 0.3

m

1.11.2 Consequen es of the dB s ale

Using the dB s ale signies that the range of hearing is transformed to sound pressure levels between

0 and 120 dB. A onstant dB step orresponds to a onstant variation in sensation. Furthermore, a

multipli ation of physi al quantities be omes a simple summation in the dB domain.

1.11.3 Subtlety of the dB s ale

The question of the relevan e of a hange of x dB an be answered for example with help of the human

auditory sensation a ording to Table 1.3.

hange in sound pressure level sensation

< 2 dB not audible

2. . .4 dB just audible

5. . .10 dB learly audible

> 10 dB very onvin ing

Table 1.3: Sensation for hanges in sound pressure level for the human hearing.

An other hint regarding the relevan e of level dieren es an be derived from toleran es of modern

sound level meters. The overall un ertainty of su h devi es in the order of 1 dB.

1.11.4 Computations involving dB quantities

Spe ial are is needed when al ulations involve dB quantities. The addition of dB values orresponds

to a multipli ation of the physi al quantities. Very often a summation of physi al quantities is needed.

In this ase the dB values rstly have to be onverted ba k to linear quantities before the operation an

be applied. It has to be onsidered, what quantities add up. In ase of oherent signals the summation

goes for sound pressure, in ase of in oherent ontributions the orresponding sound pressure square

values have to be summed up. Often the result is then again onverted and expressed as a level.

1.11.5 Typi al values of sound pressure levels

Table 1.4 indi ates typi al sound pressure levels at a ertain distan e for dierent sound sour es.

sound sour e sound pressure level

ti k of an alarm lo k in 0.5 m 30 dB

human voi e in 2 m 60 dB

road tra in 10 m (1000 vehi les/h, 80 km/h) 70 dB

jet air plane in 100 m 120 dB

Table 1.4: Some typi al sound pressure level values.

1.12 Classi ation of a ousti al signals

A ousti al signals an be subdivided into few fundamental types. In daily life they almost never

o ur in pure form, but often one or the other fundamental type an be identied as predominant.

The following gures show the basi signal types (on the left: time dependen y, on the right: spe trum).

22

pure tone (Reinton): time dependen y pure tone: spe trum

Zeit [sec]

Sig

nal

0 0.005 0.01 0.015 0.02

Frequenz [Hz]

Am

plitu

de

0 2000 4000 6000

omplex tonal sound (Klang): time depen-

den y

omplex tonal sound: spe trum

Zeit [sec]

Sig

nal

0 0.005 0.01 0.015 0.02

Frequenz [Hz]A

mp

litu

de

0 2000 4000 6000

white noise (weisses Raus hen): time depen-

den y

white noise: spe trum

Zeit [sec]

Sig

nal

0 0.005 0.01 0.015 0.02

Frequenz [Hz]

Am

p.d

ich

te

0 2000 4000 6000

pink noise ( rosa Raus hen): time dependen y pink noise: spe trum

Zeit [sec]

Sig

nal

0 0.005 0.01 0.015 0.02

Frequenz [Hz]

Am

p.d

ich

te

0 2000 4000 6000

f-0.5

500 Hz o tave band ltered noise: time de-

penden y

500 Hz o tave band ltered noise: spe trum

Zeit [sec]

Sig

nal

0 0.005 0.01 0.015 0.02

Frequenz [Hz]

log

. A

mp

.dic

hte

100 1000 10000

23

500 Hz third o tave band ltered noise: time

dependen y

500 Hz third o tave band ltered noise: spe -

trum

Z eit [s ec]

Sig

nal

0 0.005 0.01 0.015 0.02

Frequenz [Hz]

log

. A

mp

.dic

hte

100 1000 10000

bang (Knall): time dependen y bang: spe trum

Zeit [sec]

Sig

nal

0 0.005 0.01 0.015 0.02

Frequenz [Hz]lo

g. A

mp

.dic

hte

0 2000 4000 6000

tone burst: time dependen y tone burst: spe trum

0 0.01 0.02 0.03Zeit [sec]

T

0 500 1000 1500 2000

Frequenz [Hz]

log

.Am

p.d

ich

te

B = 1/T

sweep: time dependen y sweep: The spe trum depends on the time

dependen y of the frequen y variation.

Zeit [sec]

Sig

nal

0 0.02 0.04 0.06 0.08

1.13 Standing waves

1.13.1 Superposition of waves traveling in opposite dire tions

The superposition of two sine waves of equal frequen y and amplitude but opposite dire tions results

in a standing wave. For a mathemati al investigation the two waves are introdu ed in omplex writing:

p1(x, t) = pej(ωt−kx) (1.103)

24

p2(x, t) = pej(ωt+kx) (1.104)

The sum yields:

ptot(x, t) = p

1(x, t) + p

2(x, t) = pejωt

(

e−jkx + ejkx)

= pejωt (cos(−kx) + j sin(−kx) + cos(kx) + j sin(kx)) = pejωt2 cos(kx) (1.105)

The superposition of the two waves is no longer a propagating wave but a harmoni os illation that is

modulated in spa e with cos(kx). As a onsequen e at ertain lo ations maxima and at other lo ations

minima arise.

1.13.2 Quarter wave length resonators

An example for the appli ation of standing waves is the quarter wave length (λ/4) resonator. It onsistsof a tube with an a ousti ally hard termination at one end. At the end with the hard termination the

sound waves are perfe tly ree ted. If the sound wave length equals four times the length of the tube,

a sound pressure minimum results at the open end of the tube. This is in oni t with sound pressure

of the ex itation outside the tube. The tube has to rea t with a high ampli ation resulting in high

pressures at the terminated end. If a pressure mi rophone is pla ed at this position its sensitivity an

easily be in reased by more than 20 dB for the resonan e frequen ies fres,i a ording to Eq. 1.106.

fres,i =2i− 1

4

c

L(1.106)

where

i: 1,2,3,...c: speed of sound

L: length of the tube

Fig. 1.19 shows an example of a measured frequen y response for a mi rophone pla ed at the end of

su h a tube.

0

10

20

30

40

50

60

0 100 200 300 400 500 600 700 800 900 1000

frequency [Hz]

am

pli

fica

tio

n [

dB

]

Figure 1.19: Measured frequen y response at the losed end of a tube of length 66 m relative to

a mi rophone pla ement in free eld. In the experiment the distan e between the sour e and the

mi rophone was 130 m. At the resonan e frequen ies (129 Hz, 387 Hz, ...) the tube produ es very

high ampli ations. Besides the resonan es a small ampli ation of typi ally 6 dB an be observed due

to the fa t that the tube a ts as a sonde mi rophone. Indeed the sound eld is observed at the lo ation

of the open end of the tube whi h was approximately in half the distan e ompared to the free eld

referen e measurement.

25

1.14 Sound eld al ulations

Generally sound eld al ulations are seeking for spa e-time dependen ies of the sound eld variables

of interest. These solutions have to fulll the wave equation and the situation spe i boundary

onditions. The boundary onditions are dened by the sour e and the presen e of possible surfa es

with their orresponding impedan e. Often the solutions are not sear hed in the time domain. In many

ases it is easier to handle the problem in the frequen y domain. To do so the task is formulated for

an arbitrary frequen y. In this ase the Helmholtz equation (1.51) an be used instead of the wave

equation. Analyti al solutions for sound elds an only be found for spe ial situations. In the general

ase, approximations or numeri al solutions based on strategies su h as Finite Elements or Boundary

Elements have to be applied.

1.14.1 General problem of ree tion

The general problem of ree tion ares about the resulting sound eld if a sound wave is ree ted at

an obje t. If the sound wave length is larger than the dimensions of the obje t the ree tion pro ess is

usually alled s attering.

In the mathemati al des ription the ree ting or s attering obje t introdu es a boundary ondition on

the obje t surfa e. This easily done for so alled lo ally rea ting surfa es whi h an be des ribed by a

surfa e impedan e (p/ ~vn). Many materials an be handled as a lo ally rea ting surfa es. On the other

hand there are stru tures that behave as extended rea ting surfa es with a relevant amount of sound

propagation in the material itself. An example for a medium with extended rea tion is ballast that is

used in the superstru ture of railway lines. Here we will restri t the dis ussion to lo ally rea ting surfa es.

In the most simple ase of an a ousti ally hard surfa e the boundary ondition simplies to

vn = 0 or

∂p

∂n= 0 (1.107)

where

vn: sound parti le velo ity omponent perpendi ular to surfa e

∂p/∂n: partial derivative of the sound pressure in dire tion perpendi ular to the surfa e

Outside the ree ting obje t the resulting sound eld has to fulll the wave equation. In the view of

the frequen y domain, the orresponding ondition is the Helmholtz equation. It is a good idea to split

up the resulting sound pressure eld p in an in ident pe and a ree ted (or s attered) ps omponent:

p = pe + ps (1.108)

Usually the in ident wave pe is known and the problem lies in the determination of the ree ted

omponent ps. The Helmholtz equation has to be fullled for the total sound eld p. With the two

parts pe and ps the Helmholtz equation be omes:

(pe + ps) + k2(pe + ps) = 0 (1.109)

Eq. 1.109 an be rewritten as

pe + k2pe +ps + k2ps = 0 (1.110)

The Helmholtz equation has also to be fullled for the in ident wave pe alone. Therefore

ps + k2ps = 0 (1.111)

It follows that the ree ted wave has to fulll the Helmholtz equation as well. In addition, the ree ted

wave has to ensure that the boundary ondition at the surfa e of the obje t is fullled. In ase of an

a ousti ally hard obje t this implies that vn = 0. With vn,e as the normal omponent of the sound

parti le velo ity of the in ident wave and vn,s as the normal omponent of the sound parti le velo ity

of the ree ted wave follows vn,s = −vn,e. This ondition an be formulated for the sound pressure

with help of Eq. 1.12:

∂ps∂n

= jωρvn,e (1.112)

26

1.14.2 Kir hho-Helmholtz Integral and Boundary Element Method

With help of Green's theorem the Helmholtz equation (1.51) an be transformed into an integral

equation (1.113), alled the Kir hho - Helmholtz integral.

p(x, y, z, ω) =1

4π

∫

S

(

jωρvS(ω)e−jωr/c

r+ pS(ω)

∂

∂n

e−jωr/c

r

)

dS (1.113)

What Eq. 1.113 says is that the omplex amplitude fun tion p at any point P in spa e an be

al ulated if the normal omponent of the sound parti le velo ity vS and the sound pressure pS is

known on a losed surfa e S. The point P an be lo ated inside or outside of S. r is the distan e

between P and the surfa e element dS, ∂/∂n stands for the derivative in the dire tion perpendi ular

to the surfa e

3

. The surfa e S may lie partially in the innity. Furthermore it an be shown that the

Kir hho Helmholtz integral holds even for points P on the surfa e S itself. Though in this ase Eq.

1.113 yields p/2.

With help of the Kir hho Helmholtz integral (KHI), typi al radiation problems an be solved for

vibrating bodies that are a ousti ally hard. It is usually assumed that the me hani al vibration is

known on the surfa e of the body. With this knowledge the problem is ompletely spe ied.

The sound parti le velo ity on the surfa e orresponds to the normal omponent of the speed of the

me hani al vibration. With this information the rst term of the KHI is known. The sound pressure on

the body surfa e as the se ond eld variable ne essary to evaluate the KHI is unknown at this point.

However it is possible to express the sound pressure by the KHI itself. By dis retization of the body

surfa e, a nite number of pressure variables an be introdu ed. This dis retization has to be ne

ompared to the shortest wave length of interest. Typi ally 6 to 10 points per wave length have to

be hosen. With the pressure variables and the KHI, a system of equations an be set up and solved.

On e the pressure is known on the surfa e, the KHI allows for the al ulation of the sound pressure

at any point in spa e outside the vibrating body. The numeri al implementation of this pro edure is

alled Boundary E lement Method or short BEM

4

.

1.14.3 Appli ations of the Kir hho-Helmholtz Integral

Rayleigh Integral for the sound radiation of a piston in an innitely extended wall

In the following it is assumed that an innitely extended and a ousti ally hard wall at rest ontains a

limited region S with given normal omponent of the velo ity vn(x, y). As the wall a ts as a ree tor

the problem an be transformed into an equivalent one with eliminated wall but an additional mirror

sour e. The mirror sour e makes sure that the resulting normal omponent of the sound parti le

velo ity omponent vanishes on the wall outside S. The translational ba k and forth movement of the

region S has thus to be repla ed by a pulsating movement where the front and ba k side both move in

phase outwards and inwards. S an be interpreted as a body with variable thi kness mounted in free

spa e (Fig. 1.20).

The sound pressure on the surfa e of the piston is unknown but due to symmetry, the values are

identi al on both sides. In the evaluation of the Kir hho-Helmholtz integral the ontribution of the

sound pressure is multiplied with the derivative of a distan e fun tion in the outward dire tion. Therefore

the pressure ontributions add up to 0 and an thus be omitted. On the other hand, the ontribution

of the velo ity is identi al for both sides of the piston S. It is su ient to perform the integration over

one side only and multiply the result by 2. The remaining relation is alled Rayleigh integral:

p(x, y, z, ω) =jωρ

2π

∫

S

vn(x, y, ω)e−jkr

rdS (1.114)

3

The derivative of the fun tion f in a point ~a in dire tion ~n is given by: limβ→0f(~a+β~n)−f(~a)

βwhere ~n is a ve tor

of length 1.

4

O hmann M., Wellner F. Bere hnung der S hallabstrahlung dreidimensionaler s hwingender Körper mit Hilfe eines

Randelemente-Mehrgitterverfahrens. A usti a 73 (1991) pp 177 - 190.

27

x

y

vn(x,y)

d/dn'd/dn''

P

r'

r''

d/dn'' f(r'') = - d/dn' f(r')

Figure 1.20: Situation of sound radiation of a piston in an innitely extended wall.

Kir hho's approximation to handle dira tion problems

The situation to dis uss here is a innitely extended s reen with an opening S and an in ident plane wave

(Fig. 1.21). If the normal omponent of the sound parti le velo ity vn(x, y) over S is known, the sound

eld behind the s reen an be al ulated by applying the Rayleigh integral. Kir hho's approximation

assumes that the sound parti le velo ity in the opening is identi al to the situation without s reen.

This simpli ation be omes more and more riti al as the opening gets smaller ompared to the wave

length. Figure 1.22 shows the al ulated sound pressure eld behind an opening using Kir hho's

approximation.

Figure 1.21: Situation of a plane wave in ident on a s reen with an opening.

Kir hho's approximation to handle problems of ree tions at small s reens

The wave that is ree ted at an a ousti ally hard s reen an be determined with help of Kir hho's

approximation as well. The s reen introdu es the boundary ondition of vanishing resulting normal

omponent of the sound parti le velo ity. Kir hho's approximation for the sound pressure lies in

the assumption that the pressure doubles in front of the s reen and vanishes at the ba k side. The

ree ted sound wave has to ensure that these onditions are fullled.

This is a omplished by a ree ted sound parti le velo ity distribution on the s reen that has equal

amplitude but opposite dire tion ompared to the in ident wave. Kir hho's approximation ignores

boundary ee ts. It is further assumed that the sound parti le velo ity is homogeneous over the s reen.

The sound pressure of the ree tion ontribution has to be identi al in amplitude and phase to the

in ident sound pressure on the front side of the s reen. On the rear side, the ree ted sound pressure

has equal amplitude but opposite phase.

Knowing sound parti le velo ity and sound pressure of the ree tion on the surfa e of the s reen, the

ree ted sound eld an be al ulated at any point by evaluating the Kir hho-Helmholtz integral

over the front and ba k side of the s reen.

28

Figure 1.22: Sound pressure eld behind a s reen with an opening of diameter 25 m. The al ulation

assumes an in ident plane wave from left to right and Kir hho's approximation. Shown are the

frequen ies 500, 1000, 2000 and 4000 Hz. The sound pressure is olor oded. Compared to the sound

pressure of the in ident wave the ampli ation due to fo using ee ts in hot spots an rea h +6 dB.

Figure 1.23 shows the situation for an in ident plane wave with sound pressure p and sound parti le

velo ity v. With vSSv and vSSr and pSSv and pSSr as sound parti le velo ities and sound pressures of

the ree ted wave on the front (v) and the rear (r) side of the s reen it an be written:

vSSv = v

vSSr = −vpSSv = p

pSSr = −p (1.115)

With the Kir hho-Helmholtz integral the sound pressure of the ree ted wave is given as

pstreu(x, y, z, ω) =1

4π

∫

S

(

jωρvSS(ω)e−jωr/c

r+ pSS(ω)

∂

∂n

e−jωr/c

r

)

dS (1.116)

As vSSv and vSSr have identi al amplitude and opposite sign, their ontributions an el ea h other

during integration over the front and rear side of the s reen. Sound pressure behaves dierently. The

derivative of the distan e fun tion yields opposite signs for the front and rear side. As pSSv and pSSrhave opposite signs as well they add up onstru tively. Instead of integrating over the front and the

rear side, the integration an be restri ted to the front side and the result is multiplied with a fa tor of

2. Figure 1.24 shows the result of su h a al ulation.

Huygens elementary sour es and the onstru tion of Fresnel zones

As shown above the al ulation of the ree tion at a s reen or the dira tion at an opening requires

the evaluation of the Kir hho-Helmholtz or the Rayleigh integral. This integration orresponds to

29

Figure 1.23: Ree tion of a sound wave at a s reen. Note that the sound parti le velo ity normal

omponents are oriented in the outward dire tion.

-30

-25

-20

-15

-10

-5

0

31.25 62.5 125 250 500 1000 2000 4000 8000

frequency [Hz]

refl

ect

ion

rel.

dir

ect

so

un

d [

dB

]

Figure 1.24: Kir hho-Helmholtz integral al ulation of the frequen y response of the normal in ident

ree tion at a s reen of dimensions 2×2 m. The sour e is at 5 m, the re eiver at 10 m distan e to

the ree tor. The dis retization used in the al ulation was set to 1/10 of the wave length under

onsideration. At low frequen ies the ree tion is weak. A tually it is more likely a s attering in all

dire tions. At high frequen ies the sound pressure of the ree ted wave tends to -10 dB as expe ted

for the innitely extended ree tor.

a summation of the ontributions of monopole and dipole sour es on the surfa e S. In a qualitative

sense this on ept was already proposed by Huygens. He introdu ed elementary sour es on the front

of a wave to extrapolate the wave front at a later time. By this on ept, a fundamental understanding

of wave phenomena su h as dira tion an be obtained.

Fresnel developed this on ept in a more quantitative manner. In many ases the amplitude hanges

only slowly during the integration over the surfa e S. As a rst approximation it is therefore su ient

to take are of the phase hange alone. A further simpli ation is the lassi ation of the phase in just

two ategories +1 (0 . . . 180) and -1 (180 . . .360). With this the integration redu es to additions

and subtra tions of amplitudes. With the smallest phase assumed as 0, the phase lasses an be

enumerated. The phase of the nth lass lies in the interval (n − 1) × 180...n × 180. +1 - lasses

(positive ontributions) have odd n, even n stand for -1 - lasses (negative ontributions).

A region on the surfa e S for whi h the ontributions belong to the nth lass, is alled the nthFresnel zone

5

. The dimension of a Fresnel zone is frequen y dependent. For low frequen ies the

Fresnel zones are large, for high frequen ies the zones are small. On a plane surfa e the Fresnel

5

Cremer L., Fresnels Methoden zur Bere hnung von Beugungsfeldern, A usti a, vol. 72, p.1-6 (1990).

30

zones are ellipti rings. The sound pressure at a re eiver point is given as sum and dieren es of

ontributions that are proportional to the area of the Fresnel zones and inversely proportional to

the distan e. Thereby most of the ontributions do an el ea h other. Finally what remains is the

ontribution of half of the rst zone. Consequently for a plane sound wave that passes a s reen with an

opening that orresponds to the rst Fresnel zone, an ampli ation by a fa tor of 2 or +6 dB is expe ted.

The rst Fresnel zone an be regarded as the relevant region for a ree tion. If the ree tor is smaller

than half of the rst Fresnel zone, the amplitude of the ree ted sound pressure s ales with the or-

responding area ratio. Sometimes the rst Fresnel zone is dened as the region for a maximum phase

shift of only a quarter of a wave length. This automati ally a ounts for the fa t that only half of the

area of the rst λ/2-Fresnel zone remains as net ontribution for a total ree tion.

Interpretation of the Kir hho-Helmholtz integral with monopole and dipole sour es

The Kir hho-Helmholtz integral

p(x, y, z, ω) =1

4π

∫

S

(

jωρvS(ω)e−jωr/c

r+ pS(ω)

∂

∂n

e−jωr/c

r

)

dS (1.117)

an be rewritten with the following onsiderations:

k =2π

λ=ω

c⇒ e−jωr/c = e−jkr (1.118)

∂

∂n

(

e−jkr

r

)

=1

r2

(

r∂

∂n

(

e−jkr)

− e−jkr∂r

∂n

)

=1

r2

(

−jkr ∂r∂n

e−jkr − e−jkr∂r

∂n

)

=1

r2e−jkr(−jkr − 1)

∂r

∂n(1.119)

In (1.119) ∂r/∂n orresponds to the proje tion of r on the normal dire tion ~n. So we get

∂r/∂n = − cosφ where φ is the angle between the normal dire tion ~n and the dire tion to the re eiver

point (x, y, z).

Therewith the Kir hho-Helmholtz integral be omes:

p(x, y, z, ω) =1

4π

∫

S

(

jωρvS(ω)e−jkr

r+ pS(ω)

1 + jkr

r2cosφe−jkr

)

dS (1.120)

The integrand in (1.120) is omposed of two parts. The rst term orresponds to the sound pressure

produ ed by an omnidire tional monopole sour e. The se ond term is the ontribution of a dipole

sour e with the dipole axis pointing in the surfa e normal dire tion. The Kir hho-Helmholtz integral

an thus be interpreted as summation of monopole and dipole ontributions distributed over the surfa e

S. The strength of the monopole sour es is given by the normal omponent of the surfa e sound parti le

velo ity, the strength of the dipole sour es is determined by the sound pressure on S. Eq. 1.120 is

the mathemati al basis to synthesize a three dimensional sound eld by ontrolling sound pressure and

sound parti le velo ity on a losed surfa e

6

,

7

.

1.14.4 Method of Finite Dieren es

The method of nite dieren es is a widely used approa h to numeri ally solve dierential equations.

For sound eld al ulations the region of interest has to be dis retized su iently ne and represented

by a nite number of grid points. The relevant dierential equations are then approximated by linear

equations for the eld variables in the grid points. Thereby derivatives translate into dieren es.

6

A. J. Berkhout, A Holographi Approa h to A ousti al Control, Journal of the Audio Engineering So iety, vol. 36,

n.12, p.977-995 (1988).

7

Diemer de Vries, Sound Reinfor ement by Waveeld Synthesis: Adaption of the Synthesis Operator to the Loudspeaker

Dire tivity Chara teristi s, Journal of the Audio Engineering So iety, vol. 44, n.12, p.1120-1131 (1996).

31

Finite dieren es in the frequen y domain

The sound eld al ulation in the frequen y domain is usually based on the Helmholtz equation

8

.

The system of equations that has to be established uses the unknown amplitudes and phase values of

the sound eld variable (usually sound pressure) in ea h grid point. The parameters of the equations

are determined by appli ation of the Helmholtz equation and the boundary onditions. In most ases

the information about the boundary is given in form of impedan es (ratio of sound pressure and

sound parti le velo ity). As sound pressure and sound parti le velo ity are related to ea h other by a

dierential equation, it is possible to get rid of one variable in order to des ribe the boundary ondition

with one eld variable alone.

In omplex writing sound pressure and sound parti le velo ity read as

p = pejωt (1.121)

v = vejωt (1.122)

where p and v represent omplex amplitude fun tions. The impedan e Z is

Z =p

v(1.123)

With (1.12) one gets for one dire tion (here: x)

∂p

∂xejωt = −ρjωvejωt (1.124)

∂p

∂x= −ρjω p

Z(1.125)

The angular frequen y ω an be expressed with the wave number k as:

ω = 2πc

λ= kc (1.126)

By inserting Eq. (1.126) in (1.125) one nally gets

∂p

∂x= −ρcjk p

Z(1.127)

Eq. (1.127) represents an impedan e boundary ondition with only sound pressure as variable. With

this the system of linear equations for the n omplex pressure values (amplitude and phase) an be

established. In general this makes it ne essary to invert an n × n matrix for ea h frequen y. Taking

into a ount that the grid spa ing has to be in the order of 1/6 of the shortest wavelength of interest,

it be omes lear that the method is restri ted to small volumes or low frequen ies. The method is not

very exible as only homogeneous and equidistant grids an be applied.

Finite dieren es in the time domain

It is possible and often bene ial to apply the nite dieren es on ept in the time domain

9

,

10

. The

result of su h a simulation is an impulse response that ontains information about all frequen ies. An

other advantage is that fa t that an iterative, time-step wise updating s heme an be used without the

ne essity of solving a system of equations. However a di ulty with the time domain formulation is the

spe i ation of boundary onditions. Typi ally these are dened as impedan es in the frequen y domain.

An exa t transformation to the time domain would require a onvolution operation whi h is very ex-

pensive in the sense of omputational eort. Therefore appropriate approximation are usually used

11

,

12

.

8

Alfredson R.J., A Note on the Use of the Finite Dieren e Method for Predi ting Steady State Sound Fields. A usti a

28 (1973) pp 296 - 301.

9

D. Botteldooren, Finite-dieren e time-domain simulation of low-frequen y room a ousti problems, Journal of the

A ousti al So iety of Ameri a, vol. 98, p.3302-3308 (1995).

10

S. Sakamoto, H. Ta hibana, Numeri al study on sound propagation from depressed/semi-underground roads, Pro-

eedings inter-noise 2001.

11

B. Van den Nieuwenhof, J.-P. Coyette, Treatment of frequen y-dependent admittan e boundary onditions in transient

a ousti nite/innite-element models, Journal of the A ousti al So iety of Ameri a, vol. 110, p.1743-1751 (2001)

12

Benoit Van den Nieuwenhof, Jean-Pierre Coyette, Treatment of frequen y-dependent admittan e boundary onditions

in transient a ousti nite-innite-element models, Journal of the A ousti al So iety of Ameri a, vol. 111, p.1743-1751

(2001).

32

The method of F inite Dieren es in the T ime Domain (FDTD) is based on Eq. 1.12 and 1.29. In

artesian oordinates these equations read as:

∂p

∂x= −ρ∂vx

∂t(1.128)

∂p

∂y= −ρ∂vy

∂t(1.129)

∂p

∂z= −ρ∂vz

∂t(1.130)

−∂p∂t

= κP0

(

∂vx∂x

+∂vy∂y

+∂vz∂z

)

(1.131)

The region of interest is overed with a regular grid. The sound pressure is evaluated at grid positions

< i∆x, j∆y, k∆z > where i, j, k are whole-numbered indi es and ∆x,∆y,∆z are the dis retization

widths in the three oordinate dire tions. The sound parti le velo ity omponent in the x-dire tion is

evaluated at grid points < (i± 0.5)∆x, j∆y, k∆z >, the y- omponent at < i∆x, (j ± 0.5)∆y, k∆z >and the z- omponent at < i∆x, j∆y, (k±0.5)∆z >. The Figure 1.25 shows the grid in two dimensions.

vx

vy

ii-1 i+1

j

j-1

j+1

i-0.5 i+0.5

j+0.5

j-0.5

p vx

vy

Figure 1.25: FDTD dis retization of the region of simulation in the two dimensional ase. Sound

pressure is investigated at < i∆x, j∆y >, the sound parti le velo ity omponents in the x and the

y-dire tion at < (i± 0.5)∆x, j∆y > and < i∆x, (j ± 0.5)∆y >. The dashed line marks the border of

the region where boundary onditions have to be dened.

The original dierential equations of the sound eld are approximated by nite dieren es. Besides a

spa ial dis retization, a temporal dis retization has to be introdu ed for that purpose. Sound parti le

velo ity is evaluated at times t = (l + 0.5)∆t, sound pressure is evaluated at times t = l∆t (l being a

running index). The orresponding dieren e equations be ome:

v[l+0.5]x (i+ 0.5, j, k) = v[l−0.5]

x (i + 0.5, j, k)− ∆t

ρ∆x

(

p[l](i + 1, j, k)− p[l](i, j, k))

(1.132)

v[l+0.5]y (i, j + 0.5, k) = v[l−0.5]

y (i, j + 0.5, k)− ∆t

ρ∆y

(

p[l](i, j + 1, k)− p[l](i, j, k))

(1.133)

v[l+0.5]z (i, j, k + 0.5) = v[l−0.5]

z (i, j, k + 0.5)− ∆t

ρ∆z

(

p[l](i, j, k + 1)− p[l](i, j, k))

(1.134)

p[l+1](i, j, k) = p[l](i, j, k)− ρc2∆t

∆x

(

v[l+0.5]x (i+ 0.5, j, k)− v[l+0.5]

x (i− 0.5, j, k))

−ρc2∆t

∆y

(

v[l+0.5]y (i, j + 0.5, k)− v[l+0.5]

y (i, j − 0.5, k))

−ρc2∆t

∆z

(

v[l+0.5]z (i, j, k + 0.5)− v[l+0.5]

z (i, j, k − 0.5))

(1.135)

33

For points at the border of the simulation region, boundary onditions have to be dened. A di ulty

is the handling of an open spa e. At the border of the region of al ulation, total ree tion o urs.

To avoid unwanted artifa ts, a zone with damped propagation has to be introdu ed. A very e ient

method is the perfe tly mat hed layer, originally proposed for ele tro-magneti eld al ulations

13

. Without signi ant restri tions it an be assumed that the boundary onditions need only be

formulated at grid points where the sound parti le velo ity is evaluated. A lo al rea tion ondition

is usually assumed whi h means that the boundary onditions makes a statement about the ratio

of sound pressure and the normal omponent of the sound parti le velo ity. This orresponds to an

impedan e that is usually frequen y dependent.

If the possible frequen y dependen y is restri ted, the formulation of the boundary onditions simplies

dramati ally. Here it is assumed that the impedan e an be des ribed with Eq. 1.136. In

14

a more

subtle se ond order extension is dis ussed.

Z(ω) = a−11

jω+ a0 + a1jω (1.136)

where

a−1, a0, a1: positive real numbers.

For the Fourier transform in the frequen y domain it an be written:

P (ω) = Z(ω)V (ω) = V (ω)a−11

jω+ V (ω)a0 + V (ω)a1jω (1.137)

where

P (ω): Fourier transform of the sound pressure time history

V (ω): Fourier transform of the sound parti le velo ity time history

Equation 1.137 translates into the time domain as:

p(t) =

∫ t

−∞a−1vn(τ)dτ + a0vn(t) + a1

dvn(t)

dt(1.138)

As already mentioned it is assumed that the boundary ondition is dened at a grid point where the

sound parti le velo ity is evaluated. In these points Eq. 1.132 to 1.134 have to be repla ed a ordingly.

Exemplarily this is demonstrated here for the sound parti le velo ity omponent in the x-dire tionwith the assumption, that the border runs through the grid point < (i+ 0.5)∆x, j∆y, k∆z > and the

simulation region lies on the left (at lower x values).

As for any point in spa e, Eq. 1.139 has to be fulllled for the boundary point < (i +0.5)∆x, j∆y, k∆z > as well.

∂p

∂x= −ρ∂vx

∂t(1.139)

In ontrast to the symmetri al approximation from above (1.132), a onesided approximation for (1.139)

is used here:

v[l+0.5]x (i+ 0.5, j, k) = v[l−0.5]

x (i+ 0.5, j, k)− 2∆t

ρ∆x

(

p[l](i+ 0.5, j, k)− p[l](i, j, k))

(1.140)

In Eq. 1.140 the sound pressure at the point < (i + 0.5)∆x, j∆y, k∆z > is unknown. However with

knowledge of the boundary ondition (1.138) this unknown sound pressure an be expressed with the

sound parti le velo ity:

13

J. P. Berenger, A perfe tly mat hed layer for the absorption of ele tro magneti waves, Journal of Computational

Physi s, vol. 114, p.185-200 (1994)

14

K. Heuts hi, M. Horvath, J. Hofmann, Simulation of Ground Impedan e in Finite Dieren e Time Domain Cal ulations

of Outdoor Sound Propagation, A ta A usti a united with A usti a, vol. 91, 35-40 (2005).

34

p[l](i+ 0.5, j, k) = a−1∆t

(

l∑

m=−∞

v[m−0.5]x (i + 0.5, j, k)

)

+ a0v[l]x (i+ 0.5, j, k) +

+a1v[l+0.5]x (i+ 0.5, j, k)− v

[l−0.5]x (i+ 0.5, j, k)

∆t(1.141)

In Eq. 1.141 a Term with the sound parti le velo ity at time l o urs. As sound parti le velo ity values

are evaluated at times .+0.5, this value has to approximated as average of the two temporal neighbors:

v[l]x =v[l+0.5]x + v

[l−0.5]x

2(1.142)

Finally Eq. 1.140 an be dissolved for the wanted value v[l+0.5]x (i + 0.5, j, k) by insertion of Eq. 1.141

and 1.142. Eq. 1.143 is the repla ement of Eq. 1.132 as updating equation.

v[l+0.5]x (i+ 0.5, j, k) = v[l−0.5]

x (i + 0.5, j, k)ρ∆x− a0∆t+ 2a1ρ∆x+ a0∆t+ 2a1

+

+p[l](i, j, k)2∆t

ρ∆x+ a0∆t+ 2a1−

l∑

m=−∞

v[m−0.5]x (i+ 0.5, j, k)

2a−1(∆t)2

ρ∆x+ a0∆t+ 2a1(1.143)

The innite sum in Eq. 1.143 makes it ne essary to introdu e an additional register to a umulate the

orresponding ontributions over time.

If the region of simulation lies on the right side of the boundary the updating equation for the sound

parti le velo ity is found as:

v[l+0.5]x (i + 0.5, j, k) = v[l−0.5]

x (i+ 0.5, j, k)ρ∆x− a0∆t+ 2a1ρ∆x+ a0∆t+ 2a1

−

−p[l](i+ 1, j, k)2∆t

ρ∆x+ a0∆t+ 2a1−

l∑

m=−∞

v[m−0.5]x (i + 0.5, j, k)

2a−1(∆t)2

ρ∆x+ a0∆t+ 2a1(1.144)

The equations for the two other oordinate dire tions are found by adapting the orresponding indi es.

A possible initial ondition to investigate the impulse response is a smooth and ontinuous sound

pressure distribution at and around the sour e position as given in Eq. 1.145. It has to be ensured that

no aliasing o urs, neither in spa e nor in time.

p[0](i, j, k) = e−((0.3(i−iS))2+(0.3(j−jS))2+(0.3(k−kS))2)(1.145)

with

iS, jS , kS : indi es of the grid point of the sour e position.

Finally the set of dieren e equations represents an updating s heme to determine new sound parti le

velo ity and sound pressure values from the orresponding old ones. Observing the rea tion on an

impulse ex itation, the temporal evolution of the sound eld at ea h grid point is obtained. These

impulse responses represent the omplete information about the system. By appli ation of a Fourier

transformation the frequen y responses an easily be al ulated.

Figure 1.26 shows an example of a FDTD simulation.

1.14.5 Method of nite elements

As in many dis iplines Finite Elements an su essfully be applied for sound eld al ulations

15

,

16

. The nite element method is espe ially well suited for bounded domains, however it is

15

W. J. Anderson, Numeri al A ousti s, Multimedia study guide (CD-ROM), Ann Arbor, Automated Analysis Corpo-

ration (1996).

16

G. Dhatt, G. Touzot, The Finite Element Method Displayed, John Wiley & Sons (1984).

35

Figure 1.26: 2D FDTD simulation of the temporal evolution of the sound eld in a road gallery ( ross

se tional view) after ex itation with a pressure pulse. Dark red orresponds to high positive, dark blue

to high negative sound pressure.

possible to handle innite domains as well with help of so alled innite elements. The underlying

equations are usually formulated in the frequen y domain, but time domain approa hes are also possible.

In the following the prin iples of the nite element method are introdu ed for a general 3-dimensional

36

bounded domain. The sound eld variable of interest is usually sound pressure. It is assumed that the

time dependen y is sinusoidal with angular frequen y ω. Consequently the sear h of the sound eld

redu es to the determination of the the omplex amplitude p as a fun tion of the lo ation.

The general problem an be formulated by the Helmholtz equation and three possible boundary ondi-

tions as follows:

∇2p+ k2p = 0 in the onsidered volume V (1.146)

p = p on the surfa e S1 (1.147)

vn = vn → ∂p

∂n= −jρωvn on the surfa e S2 (1.148)

vnp

= An =1

Zn→ ∂p

∂n= −jρωAnp on the surfa e S3 (1.149)

where

k: wave number =

ωc

p: predened sound pressure

vn: predened normal omponent of the sound parti le velo ity

An: predened admittan e

Zn: predened impedan e

S1, S2 and S3 form the total surfa e S that en loses the eld volume ompletely. With the nite

element pro edure an approximate solution p′ for the true pressure p is sear hed. The quality of the

approximation is measured with help of the residues that orrespond to the dieren es between the

a tual and the nominal values:

RV = ∇2p′ + k2p′ (1.150)

RS1 = p− p′ (1.151)

RS2 = −∂p′

∂n− jρωvn (1.152)

RS3 = −∂p′

∂n− jρωAnp

′(1.153)

where

RV : residuum for the onsidered volume VRS1

: residuum for the surfa e S1 with predened sound pressure pRS2

: residuum for the surfa e S2 with predened normal omponent of the sound parti le velo ity vnRS3

: residuum for the surfa e S3 with predened admittan e An

The approximate solution p′ is sear hed for the ondition of a vanishing weighted average sum of the

residues:

∫

V

WRV dV +

∫

S1

WRS1dS +

∫

S2

WRS2dS +

∫

S3

WRS3dS = 0 (1.154)

with

W : weighting fun tion

The weighting fun tion W in Eq. 1.154 an be hosen arbitrarily. However the solution p′, that fulllsEq. 1.154 depends on W . On the surfa e S1 the boundary ondition spe ies the sound pressure p.It is most plausible to hoose there p′ identi al to p. Consequently on S1 the residuum RS1

vanishes

independently of W . As will be seen later it is bene ial to hose W in su h a way that it takes the

value 0 on S1. Inserting the residues in Eq. 1.154 yields:

∫

V

W∇2p′dV +

∫

V

Wk2p′dV −∫

S2

W

(

∂p′

∂n+ jρωvn

)

dS−∫

S3

W

(

∂p′

∂n+ jρωAnp

′

)

dS = 0 (1.155)

37

The rst summand in Eq. 1.155 an be rewritten with the rst Green's formula:

∫

V

W∇2p′dV = −∫

V

gradW • gradp′dV +

∮

S

W∂p′

∂ndS (1.156)

where

•: s alar produ t

In Eq. 1.156 the integration over the surfa e S an be written as sum of the integrals over the partial

surfa es S1, S2 and S3. Finally Eq. 1.155 be omes:

−∫

V

gradW•gradp′dV +

∫

V

Wk2p′dV +

∫

S1

W∂p′

∂ndS−

∫

S2

WjρωvndS−∫

S3

WjρωAnp′dS = 0 (1.157)

In Eq. 1.157 the integration over S1 vanishes as the weighting fun tion W was hosen to 0 on S1.

The next step is the dis retization. That fore the whole region of interest is subdivided into small

elements. These elements may vary in size and may have dierent shapes (Fig. 1.27). By suitable

element sele tion, an optimal adaption to the geometry of interest is possible. This exibility is an

essential advantage ompared to the equidistant dis retisation in the nite dieren es method.

Figure 1.27: Examples of 2D and 3D nite elements.

An element des ribes a small part of the eld region of interest. In three dimensions these an be

ubes, tetrahedrons and so on. Suitable shapes in two dimensions are triangles and four-sided forms.

An element is dened by nodes that are typi ally lo ated at the orners. The elements have to over

the whole simulation region. Some elements share a ommon boundary and some have the same nodes.

For ea h element M , so alled interpolation fun tions or shape fun tions Ni are determined where M orresponds to the number of the nodes of the element. The interpolation fun tions Ni depend on

lo ation and des ribe the eld variable p′ within the element from the values at the nodes (1.158).

Outside of the element the fun tions Ni vanish.

p′(x, y, z) =

M∑

i=1

p′iNi(x, y, z) (1.158)

with

p′i: sound pressure in node iNi(x, y, z): interpolation fun tion i

The nite element algorithms dier in the hoi e of the weighting fun tions W . A ommon approa h

is the so alled Galerkin method. Thereby the weighting fun tions are identi al to the interpolation

fun tions. The formula 1.157 represents one equation for ea h element and node. These equations

ontain information about ea h isolated element only. In a so alled assembling pro edure the

equations are put together under onsideration of the fa t that some elements have ommon nodes.

This pro ess introdu es the situation geometry. In the last step the resulting system of equations has

to be solved for the eld variable sound pressure in ea h node.

As already mentioned above the nite element method is very well suited for bounded domains. Open

domains an be treated with the idea of innite elements

17

,

18

. An alternative approa h for unbounded

17

D. S. Burnett, A three-dimensional a ousti innite element based on a prolate spheroidal multipole expansion.

Journal of the A ousti al So iety of Ameri a, vol. 96, p.2798-2816 (1994).

18

D. S. Burnett, R. L. Holford, An ellipsoidal a ousti innite element. Comput. Methods Appl. Me h. Eng. vol. 164,

p.49-76 (1998).

38

domains (whi h means nothing is ree ted ba k) is the introdu tion of an arbitrary boundary where the

boundary onditions orresponds to the free eld impedan e Z = ρc. For plane waves this works ne,however in the general ase a ertain impedan e dis ontinuity will o ur, resulting in some ree ted

sound energy.

Within the on ept of nite elements it is possible to a ount for lo ally varying medium properties and

thus propagation onditions. Furthermore oupled stru ture uid systems an be treated, taking into

a ount e.g. the for e of the sound wave that is a ting on a stru ture.

1.14.6 A ousti al Holography

As already dis ussed, the Helmholtz equation an be transformed into the Kir hho-Helmholtz integral

by use of Greens theorem. For that purpose the free eld Green's fun tion is applied. The Kir hho-

Helmholtz integral expresses the sound pressure in an arbitrary point in three dimensional spa e by the

integral evaluated on a losed surfa e S. For ertain geometries of S, other Green's fun tions may be

applied that deliver simpler eld des riptions. A ase of su h a spe ially hosen surfa e is a plane that

is losed in innity in form of a hemisphere (Figure 1.28).

source

S

Figure 1.28: The surfa e S en loses the sour e ompletely. S onsists of a plane and a hemisphere

with innite radius.

Using Sommerfeld's radiation ondition, the ontribution of the integral over the hemisphere of S an

be negle ted, meaning that the integral has to be evaluated over the plane only. An adapted Green's

fun tion that takes the mirror sour e into a ount yields an integral formulation with sound pressure

alone, the ontribution of the sound parti le velo ity vanishes. The sound pressure at any point in spa e

on the right hand side of the plane (in the half spa e not o upied by the sour e) is then given as

19

:

p(x, y, z, ω) = j

∫

S

pS(ω) cosφ

(

1− j

kr

)

e−jkr

λrdS (1.159)

where

pS(ω): sound pressure (amplitude and phase) on the plane Sλ: wavelengthω: angular frequen yk: wave number = 2π/λr: distan e of the point of interest (x, y, z) to the point on the plane

φ: angle between the dire tion from the point on the plane to the point of interest and the normal

dire tion of the plane

Most remarkable in Eq. 1.159 is the fa t that a 3D sound pressure eld is determined by the sound

pressure distribution over a 2D plane. This is the essential property of holography where an interferen e

pattern in a photography an store information about a 3D obje t.

In a pra ti al appli ations of a ousti al holography sound pressure (with respe t to phase and

amplitude) is determined in a plane at dis rete grid points. The sampling region has to be large enough

so that the sound pressure outside an be negle ted. The sampling an be performed simultaneously

19

Jorgen Hald, STSF - a unique te hnique for s an-based Near-eld A ousti Holography without restri tions on

oheren e, Brüel + Kjaer Te hni al Review, no. 1, (1989).

39

with an array of mi rophones or sequentially with one mi rophone that is moved from one sampling

position to the other. In this ase a referen e is needed to determine the phase.

In some ases one is interested in the onversion of the values measured in one plane to the sound

pressure in an other plane. This operation an be performed very e iently by a spa ial Fourier

transformation

20

. This al ulation an be performed for target planes that are lose to the sour e.

By this pro edure the near eld of an extended sour e an be investigated. In su h a plane partial

sour es an be dete ted easily. Information about sound parti le velo ity an be dedu ed by using the

orresponding relation of the gradient of the sound pressure (Eq. 1.12).

1.14.7 Equivalent sour es te hnique

In some ases the method of equivalent sour es an be a very e ient strategy to nd approximate so-

lutions for sound elds dened by boundary onditions and a driving sour e. Cases with rigid boundaries

are espe ially well suited. The basi idea is to introdu e auxiliary sour es in order to satisfy the boundary

onditions. To adjust the position and strength of the auxiliary sour es an optimization pro edure is

needed. The quality of a solution is measured as the sum of the squared error at dis rete points on the

boundary. In general the error an not be made zero be ause the number of auxiliary sour es is usually

mu h lower than the number of test points on the boundary. The art in the appli ation of the method

is to nd reasonably good solutions with a low number of auxiliary sour es

21

.

1.14.8 Prin iple of re ipro ity

In a homogeneous medium at rest the so alled prin iple of re ipro ity holds for a ousti al quantities

su h as sound pressure or sound parti le velo ity

22

,

23

. The prin iple states that the ee t at a re eiver

point that is produ ed by a sour e is identi al if sour e and re eiver are ex hanged. In free eld

situations the validity of the prin iple is obvious. However the inter hangeability is maintained even if

arbitrary boundaries su h as walls and ree tors are introdu ed. In general the prin iple of re ipro ity

is violated for sound propagation outdoors due to the fa t that the medium is not at rest and not

homogeneous.

A remarkable onsequen e of this prin iple is the so alled time-reversed a ousti s.

24

,

25

. In a typi al

experiment rstly the sound emitted by a sour e is registered at several re eivers in the vi inity. Then

the re orded signals at ea h re eiver are emitted time-inverted (ba kwards) at these former re eiver

positions. In a ordan e with the prin iple of re ipro ity the emitted signals will fo us perfe tly in

the original sour e position. This fo ussing ee t is espe ially pronoun ed if sour es and re eivers are

omnidire tional.

Although experiments with time-reversed a ousti s are usually performed with several mi rophones

and onsequently several loudspeakers, the prin iple an also be applied with a single mi rophone and

loudspeaker. However in this ase ree tions are needed to produ e relevant fo using ampli ations.

It is assumed that the sour e emits a short pulse. The re eiver will then re ord the impulse response

of the system. The prin iple of re ipro ity states that this impulse response from the sour e to the

re eiver is identi al to the impulse response from the re eiver to the sour e. If the time-inverted

impulse response signal is emitted at the original re eiver position, the signal that results at the original

sour e position orresponds to the onvolution of the time-inverted impulse response with the impulse

response. This operation yields the auto orrelation fun tion of the impulse response with a distin t

peak at the orresponding point in time.

Time-reversed a ousti s an be found e.g. in medi al appli ations for diagnosis purposes and in me-

hani al treatments su h as destroying of kidney stones.

20

Maynard, J. D. et al., Neareld a ousti holography. I: Theory of generalized holography and the development of

NAH, J. A ousti al So iety of Ameri a, vol. 78, p. 1395- (1985).

21

M. E. Johnson, An equivalent sour e te hnique for al ulating the sound eld inside an en losure ontaining s attering

obje ts. Journal of the A ousti al So iety of Ameri a, vol. 104, p.1221-1231 (1998).

22

Allan D. Pier e, A ousti s, published by the A ousti al So iety of Ameri a (1989).

23

M. He kl, H.A. Müller, Tas henbu h der Te hnis hen Akustik, Springer-Verlag (1994).

24

M. Fink, Zeitumkehr-Akustik, Spektrum der Wissens haft, p.68-74, März (2000).

25

M. Fink, Time-Reversed A ousti s, Physi s Today, vol.50, p.34-40 (1997).

40

1.15 Produ tion of sound

Audible sound pressure an be understood as AC omponent of the absolute air pressure. The generation

of sound makes thus in one form or the other a time varying ex itation ne essary. Possible sound

generation me hanisms are:

• abrupt relaxation of ompressed air (bursting balloon)

• abrupt gas produ tion (explosion)

• modulated air ow (siren)

• os illating air olumn (organ pipe, a ousti al laser

26

...)

• vibrating body (loudspeaker membrane, tuning fork)

• abrupt lo al heating of air (lightening and thunder)

1.15.1 Relaxation of ompressed air

A possible sour e to produ e an impulse-like sound is a bursting balloon. The balloon lled with air

represents a volume of higher pressure. At the moment of bursting this over-pressure an propagate in

all dire tions. Thereby peak levels may ex eed 125 dB in a distan e of 1 m.

1.15.2 Abrupt gas produ tion (explosion)

The muzzle blast of a re arm is the result of an abrupt gas produ tion. An other example is the

air bag widely applied in ars. In ase of an a ident a small explosion is ignited that inates a bag

to me hani ally prote t the passenger. On the other hand the inating bag leads to very high sound

pressure peaks that may damage the ear

27

. The linear peak ranges up to 167 dB, the linear event or

exposure level is about 139 dB. These values surpass the SUVA limiting values for impulsive noise by

6 to 8 dB.

1.15.3 Modulated air ow

A modulated air ow an produ e very high sound pressure values. Probably the most ommon

appli ation of this prin iple is a siren. In its simplest form a siren onsists of a perforated rotating disk

that ontrols the passage of an air ow. The speed of revolution and the geometry of the holes in the

disk dene the frequen y of the generated sound.

An other appli ation is the so alled airow speaker. This speaker onsists of a unit that ontains

air under high pressure. By a valve that is ontrolled by the audio signal an airow produ ing very

high sound pressure an be established. A major hallenge with airow speakers is the suppression of

unwanted ow noise that appears at the nozzle.

1.15.4 Os illating air olumn

The air olumn in a tube represents a system of resonan es that an be used to generate tones. Here

the system of an organ pipe shall be dis ussed in some detail.

The organ pipe is ex ited at one end to maintain the os illation at the resonan e frequen y while the

other end is terminated by a ertain impedan e ZL. It is assumed that the tube has the length L. Theregion of interest ranges thus from x = 0 to x = L where the ex itation is at x = 0 (Figure 1.29). In

a rst step the impedan e seen at the input (x = 0) will be determined.

It is assumed that the wave length is mu h larger than the diameter of the tube. With this in mind the

sound propagation an be des ribed as an in ident and a ree ted plane wave running in x-dire tion.Sound pressure and sound parti le velo ity are in phase everywhere, their ratio orresponds to ρc.

26

E hos, The newsletter of the A ousti al So iety of Ameri a, no. 3, vol 10 (2000).

27

Beat W. Hohmann, Gehörs häden dur h Airbags, Forts hritte der Akustik DAGA 98, p.722-723 (1998).

41

x=0 x=L

ZIN

ZL

Figure 1.29: Situation for the dis ussion of the impedan es at arbitrary position in the organ tube.

Thermal and vis ous losses at the ir umferen e are ignored.

Assuming a harmoni os illation with angular frequen y ω, sound pressure and sound parti le velo ity

of the wave running to the right are given as:

pr(x, t) = Ae−jkxejωt (1.160)

vr(x, t) =A

ρce−jkxejωt (1.161)

where

A: amplitude of sound pressure

k: wave number = 2π/λ

Sound pressure and sound parti le velo ity of the wave running to the left are given as:

pl(x, t) = Bejkxejωt (1.162)

vl(x, t) = −B

ρcejkxejωt (1.163)

where

B: amplitude of sound pressure

It should be noted that the sound parti le velo ity of the wave running to the left has a negative sign.

The superposition of both waves yields the total sound eld:

p(x, t) =(

Ae−jkx +Bejkx)

ejωt (1.164)

v(x, t) =

(

A

ρce−jkx − B

ρcejkx

)

ejωt (1.165)

At the position x = L the impedan e is known, namely ZL:

p(L, t)

v(L, t)=

Ae−jkL +BejkL

Aρce

−jkL − Bρce

jkL= ZL (1.166)

From this the ree tion fa tor (the ratio of the onstants B to A) an be determined as

B

A= e−2jkLZL − ρc

ZL + ρc(1.167)

With knowledge of this ratio (Eq. 1.167) the input impedan e at the position of ex itation x = 0 an

be found as

ZIN =p(0, t)

v(0, t)= ρc

1 + BA

1− BA

= ρcZL cos(kL) + jρc sin(kL)

jZL sin(kL) + ρc cos(kL)(1.168)

Regarding the termination of the tube ZL, two important ases an be distinguished:

• losed end: ZL = ∞

• open end: ZL = radiation impedan e of the opening

42

For the losed end the input impedan e is

ZIN = −jρc cot(kL) (1.169)

For the open end and at low frequen ies k × tube diameter ≪ 1 the radiation impedan e ZL is mu h

smaller than ρc. With this follows

ZIN = jρc tan(kL) (1.170)

At higher frequen ies the radiation impedan e an no longer be negle ted. This ee t an be modeled

by an end orre tion.

The ex itation pro ess is air that is blown a ross a utting edge. This results in high sound parti le

velo ity and low sound pressure. The organ pipe is in resonan e, if this ex itation ondition is supported

by the tube, that is to say ZIN = 0. From the relations above the resonan e frequen ies an be

al ulated as:

losed end:

cot(kL) = 0 → kL = (2n− 1)π

2→ ω = (2n− 1)

πc

2L, n = 1, 2, . . .

the fundamental mode n = 1 orresponds to L =λ

4(1.171)

open end:

tan(kL) = 0 → kL = nπ → ω = nπc

L, n = 1, 2, . . .

the fundamental mode n = 1 orresponds to L =λ

2(1.172)

1.15.5 Vibrating bodies

Many sound sour es are based on vibrating bodies, su h as loudspeaker membranes, string instruments,

motors, wheels, and so forth. If the normal omponent of the surfa e velo ity is known, the sound

pressure an be al ulated at any point in spa e by appli ation of the Boundary Element method. The

required surfa e velo ity an be measured e.g. with laser vibrometers.

Strings

Vibrating strings played an important role in early history of a ousti s. Experiments with strings

allowed for the dis overy of musi al intervals and made it possible to establish a relation between the

pit h of a musi al tone and the number of os illations per se ond.

Many instruments ontain strings as ex itation element. Due to the small ross se tional dimensions,

a vibrating string is a very ine ient sound radiator. For improved radiation, the vibrations of the

strings are usually oupled to larger areas and bodies. In the following paragraph the wave equation

for the transverse motion of a string will be dedu ed.

Figure 1.30 shows a short segment of a string with the for e ve tors

~T . The amplitudes of the for e

ve tors on both sides of the segment have to be equal. If the string is not in its neutral position they

do not point exa tly in opposite dire tions. The resulting for e omponent in y dire tion a ts as a

restoring for e.

The resulting for e omponent Fres,y in y-dire tion is

Fres,y = T sin(θ(x+ dx)) − T sin(θ(x)) (1.173)

where

θ(x): angle of the string for e at position xθ(x+ dx): angle of the string for e at position x+ dx

43

x x+dx

y

y+dy

T

T

Figure 1.30: Short segment of a string with the for e ve tors T .

The fun tion sin(θ(x + dx)) an be developed as a Taylor series a ording to

f(x+ dx) = f(x) +∂f(x)

∂xdx+ . . . (1.174)

Ignoring the higher order terms, Eq. 1.173 an be written as

Fres,y = T sin(θ(x)) + T∂(sin(θ(x)))

∂xdx − T sin(θ(x)) = T

∂(sin(θ(x)))

∂xdx (1.175)

Under the assumption that the displa ement of the string is small, the angle θ remains small as well.

With this we get

sin θ ≈ tan θ ≈ ∂y

∂xfür θ → 0 (1.176)

insertion of Eq. 1.176 in Eq. 1.175 yields:

Fres,y = T∂(

∂y∂x

)

∂xdx = T

∂2y

∂x2dx (1.177)

With Newton's law the for e Fres,y in Eq. 1.177 an be expressed with mass and a eleration:

T∂2y

∂x2dx = µdxay = µdx

∂2y

∂t2(1.178)

where

T : tension of the string

µ: density of the string per unit length

dx: length of onsidered string se tion

ay: a eleration in y-dire tion

Rearranging Eq. 1.178 nally yields the dierential equation of the transverse motion of the string:

∂2y

∂x2=µ

T

∂2y

∂t2(1.179)

Eq. 1.179 has the same stru ture as the one dimensional wave equation for sound. Consequently the

general solution is given by:

y = f1(ct− x) + f2(ct+ x) (1.180)

where

c: propagation velo ity =

√

Tµ

In Eq. 1.180 f1 and f2 denote two arbitrary fun tions. The arguments (ct − x) and (ct + x) expressthat a ertain value for y an be obtained by an adjustment of time or position. This orresponds

to two waves running to the left and right. The ongurations at both ends of the string dene the

boundary onditions.

44

To nd harmoni solutions of the wave equation (1.180), the following fun tion in spa e is put on for

y:

y = A sin(ωt− kx) +B cos(ωt− kx) + C sin(ωt+ kx) +D cos(ωt+ kx) (1.181)

where

k: wave number =

ωc

For the string of length L that is lamped on both ends, the boundary onditions are:

y(0, t) = 0 und y(L, t) = 0 (1.182)

From y(0, t) = 0 follows for the parameters in Eq. 1.181: C = −A and D = −B.

Using the sum and dieren e formulas for sin(x) and cos(x):

sin(a± b) = sin(a) cos(b)± cos(a) sin(b)

cos(a∓ b) = cos(a) cos(b)± sin(a) sin(b) (1.183)

Eq. 1.181 an be simplied to

y(x, t) = −2A cos(ωt) sin(kx) + 2B sin(ωt) sin(kx) = 2 sin(kx)(B sin(ωt)−A cos(ωt)) (1.184)

The se ond ondition in Eq. 1.182 alls for sin(kL) = 0, whi h means

kL = nπ für n = 1, 2, . . . (1.185)

From that follows the ondition for the angular frequen y

ω = ncπ

L= n

√

T

µ

π

L(1.186)

The string lamped on both ends an only vibrate at dis rete frequen ies. Asso iated with ea h

frequen y is a distribution of os illation (mode) with regions of maximum and regions with minimum

os illations. However more than one mode is possible simultaneously. The o urren e of the modes

depends on the external ex itation. The most general solution of the vibrating string is the superposition

of all modes.

y(x, t) =∞∑

n=1

(An cos(ωnt) +Bn sin(ωnt)) sin(knx) (1.187)

where

ωn = n√

TµπL

kn = n πLAn, Bn: amplitude fa tor of the n-th mode, depending on the ex itation

A possible ex itation is the plu king of the string. Thereby the string is pulled away from its neutral

position at a ertain point. As a rst approximation the string forms a triangle. After the release the

string will os illate in those modes that were ex ited by this triangular shape. The orresponding modes

an be found by development of the fun tion y(x, 0) in a Fourier series. If the string is plu ked in a

distan e L/m from one end, the m-th mode is missing.

Rods

In rods dierent types of vibration an o ur:

• longitudinal (in dire tion of the rod)

• transversal (perpendi ular to the rod)

• twisting (torsion)

With ex eption of the longitudinal vibration the mathemati al des ription is expensive, see e.g.

28

.

28

Thomas D. Rossing, Neville H. Flet her, Prin iples of Vibration and Sound, Springer, 1995.

45

Membranes

Membranes are foils that are lamped at the ir umferen e. They represent so to say a two dimensional

extension of the one dimensional string. It is assumed that plane of the membrane oin ides with the

xy-plane. The dee tion from the neutral position is des ribed by the z- oordinate. Similarly to the

ase of the string, a wave equation an be formulated for the membranes, des ribing the transversal

vibration. For re tangular membranes the wave equation in artesian oordinates reads as:

∂2z

∂x2+∂2z

∂y2=σ

T

∂2z

∂t2(1.188)

where

z: dee tion of the membrane

x, y: oordinates of the membrane point

σ: density of membrane as mass per unit area

T : tension of the membrane

The above equation ignores the stiness of the membrane and the inuen e of the surrounding air.

Analogous to the string, there exist only solutions for dis rete frequen ies. These modes have to

be des ribed by a pair of nonnegative integers m,n. Figure 1.31 shows a ouple of modes for the

re tangular membrane.

m=n=1 m=2, n=1 m=1, n=2

m=n=2 m=3, n=1 m=3, n=2

Figure 1.31: Some modes for the re tangular membrane. The sides are lamped, resulting in a boundary

ondition of vanishing movement. The node lines (dashed) represent regions without movement.

For a dis ussion of ir ular membranes see the book by Rossing

29

.

1.15.6 Thermo-a ousti al ma hines

Glassblowers know the phenomenon that - under ertain ir umstan es - glass tubes an suddenly

produ e a loud pure tone when exposed to heat.

As already pointed out above, sound in air is an adiabati pro ess. This means that a passing sound

wave is a ompanied by a temperature variation, onne ted to the momentary pressure. High pressure

reates a temperature in rease while low pressure leads to a temperature de rease. Of spe ial interest

is the ase of a standing wave. Thereby air pa kages move ba k and forth. The movement in one

dire tion is onne ted to ompression and thus in reases temperature. In the other dire tion the air is

relaxing and thus ooling down. By external installation of an appropriate lo al temperature gradient

the os illation of the standing wave an be ex ited from outside.

An os illator of this type an be realized quite easily

30

. Thereby a glass tube with one open and

one losed end is used. The tube an thus a t as quarter-wave-length resonator. In the fundamental

resonan e the standing wave in the tube produ es a pressure maximum at the losed end and a

pressure minimum at the open end. Figure 1.32 shows the movement of the air parti les at progressing

moments in time.

To stimulate the resonan e situation shown in Figure 1.32 an appropriate temperature gradient

has to be established. Appropriate means that the implemented temperature gradient supports the

29

Thomas D. Rossing, Neville H. Flet her, Prin iples of Vibration and Sound, Springer, 1995.

30

Steven L. Garrett, S ott Ba khaus, The Power of Sound, Ameri an S ientist, vol. 88, no. 6, p.516-525 (2000)

46

Figure 1.32: Movie representation with progressing time from top to bottom of the movement of air

parti les in resonan e in a tube losed on the right hand end and open at the other end. While moving

to the right the air is ompressed and heated up while the movement to the left orresponds to a

relaxation with asso iated temperature de rease.

temperature gradient of the standing wave.

Instead of ex iting a sound wave by an external temperature gradient, the ee t an be reversed. If

the standing wave is ex ited by a vibrating membrane, a temperature gradient is reated by the sound

wave that be used for heating or ooling.

47

Chapter 2

A ousti al measurements

2.1 Introdu tion

A ousti al measurements an typi ally be lassied as shown in Table 2.1:

task aim example

emission measurements (passive) des ription of the strength

of a sour e

sound radiation of a lawn

mover

measurement at a re eiver position

(passive)

des ription of the strength

of a sour e in luding the

propagation to the re eiver

road tra noise measure-

ment in the living room of

a resident

measurements of a transmission sys-

tem (a tive)

des ription of a transmis-

sion system

measurement of the fre-

quen y response of a loud-

speaker

Table 2.1: Categories of typi al tasks in a ousti al measurements.

The omplete des ription of an a ousti al pro ess en ompasses the spe i ation of the time history

of sound pressure and sound parti le velo ity at ea h point. Usually for pra ti al questions one an

restri t to a few attributes of the sound eld. Most often the sound eld variable sound pressure is

investigated. Indeed sound pressure is signi antly easier to measure than sound parti le velo ity. For

sound pressure ex ellent and a urate transdu ers (mi rophones) are available to onvert the a ousti al

signal into an ele tri al one.

2.2 Signal attributes

2.2.1 Overview

As mentioned above it is usually not ne essary to represent the omplete time history of the variable of

interest. Thus the question arises what are meaningful signal attributes that an be extra ted from a

time signal. Figure 2.1 gives an example of a typi al noise-like sound pressure signal p(t). In addition

sound pressure squared p2(t) is shown.

From the time history of a signal as shown in Fig. 2.1, various attributes an be evaluated su h as:

• peak value of sound pressure or sound pressure square

• linearly or exponentially time-weighted integrations of sound pressure square

• statisti al quantities, e.g. the portion of the signal duration with sound pressure ex eeding a

ertain limit

The most ommon quantities used in a ousti al measurements are integrations over time of sound

pressure square. Peak values and statisti al quantities play only a minor role. It should be noted that the

48

0.00000 0.01000 0.02000 0.03000 0.04000 0.05000 0.06000

time [s]

−4

−3

−2

−1

0

1

2

3

4

sound p

ressure

0.00000 0.01000 0.02000 0.03000 0.04000 0.05000 0.06000

time [s]

−1

0

1

2

3

4

5

6

7

8

9

sound p

ressure

square

Figure 2.1: Typi al time history of a noise-like sound pressure signal (left) and the signal squared (right).

integration over time of sound pressure makes no sense, as this quantity yields 0 in the average. The in-

tegration of sound pressure square an be interpreted as a measure for the energy or power of the signal.

Three dierent integration quantities are used:

Momentary sound pressure level L(t):→ logarithmi form of the mean sound pressure square moving average (exponential time weighting)

L(t) = 10 log

1

RC

t∫

−∞

p2(τ)

p20e

τ−tRC dτ

[dB (2.1)

where

RC: time onstant

p(τ): instantaneous sound pressure

p0: referen e sound pressure = 2× 10−5Pa

Equivalent ontinuous sound pressure level Leq:→ logarithmi form of the mean sound pressure square taken over a ertain time frame

Leq = 10 log

1

T

T∫

0

p2(τ)

p20dτ

[dB (2.2)

where

T : measurement time interval

p(τ): instantaneous sound pressure

p0: referen e sound pressure = 2× 10−5Pa

Sound exposure level LE or SEL (former designation):

→ logarithmi form of the integral of the sound pressure square over a ertain time frame and normalized

to 1 s.

LE = 10 log

1

1 se

T∫

0

p2(τ)

p20dτ

[dB (2.3)

where

T : measurement time interval

p(τ): instantaneous sound pressure

p0: referen e sound pressure = 2× 10−5Pa

49

The momentary, moving average sound pressure level exists at any moment in time. It follows the

original signal with a more or less pronoun ed averaging ee t depending on the sele ted time onstant.

Short peaks are underestimated in their amplitude and overestimated in their pulse width. Typi al time

onstants that are FAST (125 ms) and SLOW (1 s)

1

. The time history of the FAST- or SLOW time

weighted momentary sound pressure level is typi ally evaluated for ertain single number attributes

su h as the maximum value.

The integrations over time windows of arbitrary length be ome possible when the analyzers got digital

mi ropro essors. Leq or LE both des ribe as a single value the signal power or signal energy of the

sele ted time interval.

Figure 2.2 shows the dierent integrations dis ussed above.

0.00000 0.01000 0.02000 0.03000 0.04000 0.05000 0.06000

time [s]

40

50

60

70

80

90

100

110

sound p

ressure

level [d

B]

0.00000 0.01000 0.02000 0.03000 0.04000 0.05000 0.06000

time [s]

40

50

60

70

80

90

100

110

sound p

ressure

level [d

B]

0.00000 0.01000 0.02000 0.03000 0.04000 0.05000 0.06000

time [s]

40

50

60

70

80

90

100

110

sound p

ressure

level [d

B]

0.00000 0.01000 0.02000 0.03000 0.04000 0.05000 0.06000

time [s]

40

50

60

70

80

90

100

110

sound p

ressure

level [d

B]

Figure 2.2: Momentary squared sound pressure in dB (top left), momentary sound pressure level with

small time onstant (top right), momentary sound pressure level with large time onstant (bottom left)

and equivalent ontinuous sound pressure (evaluated every 5 ms) (bottom right).

2.2.2 Appli ation of the measurement attributes

Depending on the measurement task, dierent measurement attributes are used. The following list

gives some typi al examples:

Momentary sound pressure level L:

• maximum level with time onstant FAST: Lmax, Fast → attribute to des ribe shooting noise or

the passage of road vehi les

• minimum level: Lmin→ estimation of a stationary signal with o urren e of transient unwanted

noise

Equivalent ontinuous sound pressure level Leq:

• hara terization of non-stationary sour es and signals

Sound exposure level LE :

• measurement of single events su h as e.g. train passages

1

IEC Standard 61672, Ele troa ousti s - Sound level meters, 2002-05.

50

2.2.3 Algorithm to determine the moving square average

In the analogue world the moving average a ording to Eq. 2.1 an be realized by an RC low-pass

lter. The following derivation will end up with the formula for a digital implementation.

Starting point is a time signal (e.g. sound pressure) x(t). Then the moving average x2rms(t) of x2(t)

an be determined as follows:

At time t+∆t the attribute x2rms is given as

x2rms(t+∆t) =1

RC

t+∆t∫

−∞

x2(τ)e−t+∆t−τ

RC dτ

=1

RC

t∫

−∞

x2(τ)e−t−τRC e−

∆tRC dτ +

1

RC

t+∆t∫

t

x2(τ)e−t+∆t−τ

RC dτ

= e−∆tRC x2rms(t) +

1

RC

t+∆t∫

t

x2(τ)e−t+∆t−τ

RC dτ (2.4)

For RC ≫ ∆t and t < τ < t+∆t, e−t+∆t−τ

RC an be approximated as 1. Then follows

x2rms(t+∆t) ≈ e−∆tRC x2rms(t) +

1

RC

t+∆t∫

t

x2(τ)dτ (2.5)

The integral

t+∆t∫

t

x2(τ)dτ an be approximated by the area of the re tangle ∆tx2(t+∆t):

x2rms(t+∆t) ≈ e−∆tRC x2rms(t) +

1

RC∆tx2(t+∆t) (2.6)

The exponential-fun tion e−∆tRC

an be developed into a series. Ignoring the higher order terms we get:

e−∆tRC ≈ 1− ∆t

RC(2.7)

And nally

x2rms(t+∆t) ≈(

1− ∆t

RC

)

x2rms(t) +1

RC∆tx2(t+∆t)

= x2rms(t) +x2(t+∆t)− x2rms(t)

RC∆t

(2.8)

With Eq. 2.8 the moving average at time t + ∆t is expressed as the former value at time t and a

orre tion term. ∆t an be understood as sampling interval of the digital representation of the signal.

By evaluating Eq. 2.8 the moving average an easily be updated for every new signal sample. It may

be bene ial to hose ∆t in su h a way that RC/∆t orresponds to a power of 2. In this ase the

division redu es to a simple shift operation.

2.3 Filters

Up to now it was assumed that the signal attributes are evaluated for the sound pressure time history.

However it is often of interest to take into a ount the frequen y omposition of the signal. For that

reason a ousti al measurements often use lters to apply an appropriate frequen y weighting or sele t

a limited frequen y range for the analysis. The signal attributes introdu ed above an then be applied

in the same way for ltered signals.

51

2.3.1 Weighting lters

The sensitivity of the human hearing depends strongly on frequen y. For that reason frequen y

weighting lters have been dened to simulate the frequen y response of the ear. However a serious

di ulty is the fa t, that the frequen y response of the ear depends on sound pressure level. At lower

levels the frequen y dependen y is more pronoun ed than at higher levels. For that reason several lters

were originally dened. They got the names A, B and C.

2

,

3

The A-lter was designed for low levels,

the B-lter for medium levels and the C-lter for high levels. The B-lter has disappeared ompletely.

Most often used today is the A-lter, the C-lter is applied in spe ial ases only. Evaluations performed

with the A-lter are labeled with the unit dB(A).

The transfer fun tion for the C-lter is given by

4

:

TC−Filter(s) =Ks2

(s+ ω1)2(s+ ω2)2(2.9)

where

ω1 = 1.29× 102 [rad/se

ω2 = 7.67× 104 [rad/se

With f as frequen y in Hz, the amplitude in dB of the transfer fun tion of the C-lter is

C-weighting = 20log

(

1.498× 108f2

(f2 + 20.62)(f2 + 122002)

)

(2.10)

The transfer fun tion of the A-lter orresponds to the one of the C-lter but omplemented by two

zeros at the origin and two simple poles:

TA−filter(s) =Ks4

(s+ ω1)2(s+ ω2)2(s+ ω3)(s+ ω4)(2.11)

where

ω1 = 1.29× 102 [rad/se

ω2 = 7.67× 104 [rad/se

ω3 = 6.77× 102 [rad/se

ω4 = 4.64× 103 [rad/se

The amplitude in dB of the transfer fun tion results in

A-weighting = 20log

(

1.873× 108f4

(f2 + 20.62)(f2 + 122002)√

f2 + 107.72√

f2 + 737.92

)

(2.12)

Figure 2.3 shows the amplitude responses of the A- and C-lter.

In Figure 2.4 a possible RC realization of an A-lter is depi ted. The attenuation of this lter at 1

kHz is 3.2 dB, whi h means an additional ampli ation of 3.2 dB is needed, preferably at the output

to guarantee a high-resistan e lter load.

2.3.2 Filters for frequen y analysis

The frequen y analysis pro ess evaluates signal ontributions that lie within a ertain frequen y band.

For a omplete analysis the whole frequen y range of interest is divided into a series of bands that

follow ea h other seamlessly. The signal attributes dis ussed above are then evaluated for ea h band

individually. The frequen y axis an be divided in dierent ways. For a ousti al appli ations linear and

logarithmi partitioning are very ommon. A linear partitioning results in lters of onstant absolute

bandwidth, the logarithmi partitioning orresponds to lters of onstant relative bandwidth.

2

ISO Norm 10845 A ousti s - Frequen y weighting A for noise measurements. Draft 1995.

3

ISO Norm 14938 A ousti s - Revision of B- and C-weightings and Lin-response for noise measurements. 1998.

4

IEC Standard 61672, Ele troa ousti s - Sound level meters, 2002-05.

52

100 1000 10000

frequency [Hz]

50

40

30

20

10

0

10

am

plif

icati

on

[d

B]

AC

Figure 2.3: Frequen y response of the A-(blue) and C-(purple) lter. The A-weighting shows a small

ampli ation between 1 and 6 kHz.

1k

27 n

22 n 22 n 10 n

10 k 76k8 309 k

Figure 2.4: Possible realization of the A-weighting as a passive RC lter fullling the requirements of

lass 1 a ording to IEC 61672. An additional ampli ation of 3.2 dB is needed.

Filters of onstant relative bandwidth

Filters of onstant relative bandwidth have a width B that is proportional to the enter frequen y fmof the lter. As a standardized basis a enter frequen y of 1 kHz was dened. With this the omplete

series an be developed:

B = fmg (2.13)

where

B: bandwidth, evaluated at the -3 dB points

fm: enter frequen y of the lter

g: onstant

The bandwidth is distributed logarithmi ally around the enter frequen y:

fo = fmh

fu = fm1

hfo − fu = B

g = h− 1

h(2.14)

where

fo: upper limiting frequen y

fu: lower limiting frequen y

h: onstant

53

With the ondition that the lters follow ea h other seamlessly, the n-th and the n + 1-th lter are

spe ied as:

fo,n = fu,n+1 or

fm,n+1

fm,n= h2 (2.15)

For the enter frequen y of the n-th lter follows:

fm,n = 1000(h2)n for n = . . .− 3,−2,−1, 0, 1, 2, . . . (2.16)

The most important lters of this type are o tave and third-o tave lters

5

. Third o tave bands are of

spe ial interest as this partitioning of the frequen y axis is related to human per eption ( riti al bands).

The onstants g and h have to be hosen a ording to Table 2.2.

g h h2

o tave lter 0.707 212

2

third-o tave lter 0.232 216 2

13

Table 2.2: Values of the onstants g and h for o tave and third-o tave lters.

O tave lters have a bandwidth of about 70% of the enter frequen y. The bandwidth of a third-o tave

lter is about 23% of the enter frequen y. Table 2.3 shows the standardized o tave and third o tave

lter series for the audio range from 16 Hz to 16 kHz. It should be noted that the steepness of the lters

is nite, meaning that several lters show a response even in ase of narrow band signals. Nowadays

frequen y analyzers are available that an evaluate dierent signal attributes in third o tave bands

simultaneously in real-time.

Filters of onstant absolute bandwidth

Filters with onstant absolute bandwidth have a xed bandwidth independent of the enter frequen y.

Narrow band lters with typi al bandwidths of a few Hz belong to this ategory, as well as FFT analyzers.

This sort of analysis is typi al for te hni al tasks su h as the investigation of the frequen y of a pure

tone signal omponent.

2.4 Un ertainty of measurements

A ousti al signals are often noise-like and thus have random hara ter. If only a limited time is

available, the exa t determination of the signal power or the RMS (root mean square) is impossible.

Starting point for the further dis ussion is an analog, noise-like signal. It is then assumed that a nite

number of samples are taken from the signal and based on these samples a RMS value is al ulated.

This evaluation shows a fundamental un ertainty (Figure 2.5) that depends on the time window and

the signal or analysis bandwidth as will be demonstrated below.

2.4.1 Degrees of freedom of a bandlimited random signal

A onsequen e of the frequen y limitation of a random signal is the fa t that two samples lying lose to

ea h other on the time axis are no longer statisti ally independent. The narrower the frequen y band,

the more the time between the samples has to be in reased to guarantee statisti al independen y.

A sample that is not statisti ally independent relative to the pre eding one doesn't yield relevant

information and an thus be omitted.

From a random signal u(t) of bandwidth B, n statisti ally independent samples an be taken within a

time frame T 6

:

n = 2BT (2.17)

5

IEC 61260: Ele troa ousti s - O tave-band and fra tional-o tave-band lter, 1995.

6

Jens Trampe Bro h, Prin iples of Analog and Digital Frequen y Analysis, Tapir, 1981.

54

o tave band third o tave band

fu fm fo fu fm fo11.1 12.5 14.0

11.3 16 22.6 14.3 16 18.0

17.8 20 22.4

22.3 25 28.1

22.3 31.5 44.5 28.1 31.5 35.5

35.7 40 44.9

44.6 50 56.1

44.5 63 89.1 56.1 63 70.7

71.3 80 89.8

89.0 100 112

88.4 125 177 111 125 140

143 160 180

178 200 224

177 250 354 223 250 281

281 315 353

357 400 449

354 500 707 446 500 561

561 630 707

713 800 898

707 1000 1410 890 1000 1120

1110 1250 1400

1430 1600 1800

1410 2000 2830 1780 2000 2240

2230 2500 2810

2810 3150 3530

2830 4000 5660 3570 4000 4490

4460 5000 5610

5610 6300 7070

5660 8000 11300 7130 8000 8980

8900 10000 11200

11100 12500 14000

11300 16000 22600 14300 16000 18000

17800 20000 22400

Table 2.3: Standard o tave and third o tave lters with their enter and their lower and upper limiting

frequen ies.

The variable n denotes the degrees of freedom of the signal u(t) in the time window T . Taking into

a ount that a bandlimited signal an be interpreted as an amplitude modulated arrier and that the

information is ontained in the modulation, Eq. 2.17 follows dire tly from the sampling theorem.

2.4.2 Expe tation value and varian e of various fun tions of statisti ally

independent samples

Here a gaussian random signal u(t) is assumed with expe tation value = 0 and varian e = 1 (u2rms = 1).From this signal a ertain number of statisti ally independent samples ui are taken. The set of the

samples orresponds to the random variable U .

Ampli ation

An ampli ation of the signal u(t) by a fa tor α results in an random variable U ′where

u′i = αui

expe tation value(U ′) = 0

varian e(U ′) = α2(2.18)

55

0 200 400 600 800 1000

measurement index

66

68

70

72

74

76

78

80

82

RM

S [

dB]

Figure 2.5: Spread of Leq-measurements of pink noise for an analysis bandwidth of 10 Hz and an

integration time of 0.5 s. The values lie asymmetri al relative to the true average value (red).

Summation over uiBased on the random variable U , a new quantity U ′′

is generated by summation over n samples of U .For U ′′

follows:

u′′i =

m∑

i=1

ui

expe tation value(U ′′) = 0

varian e(U ′′) = m (2.19)

Summation over u2iBased on U a new quantity U ′′′

is determined by summation over m squared values of the samples.

U ′′′is χ2

distributed with:

u′′′i =

m∑

i=1

u2i

expe tation value(U ′′′) = m

varian e(U ′′′) = 2m (2.20)

Fig. 2.6 shows the density fun tion f(y) of the χ2distribution for dierent values of the parameter m

(degrees of freedom)

7

.

2.4.3 Un ertainty of the al ulation of the root mean square

The rst step in the al ulation of the RMS (root mean square) of a signal u(t) is the determi-

nation of the available number of independent samples. For a xed time frame T and an analysis

bandwidth B this number orresponds to degrees of freedom n = 2BT a ording to Eq. 2.17. The

square of the RMS value is found as summation of the n squared samples and division by the number n.

It is assumed that the varian e of the signal under investigation u(t) equals 1. The un ertainty of the

sum S of the n samples an then be estimated by the quantiles of the orresponding χ2distribution.

The quantile χ2n,1−α is the value for S that is ex eeded with a probability α. Table 2.4 gives some

quantiles of the χ2distribution.

From Table 2.4 follows nally the un ertainty of the RMS value in the de ibel s ale. For that purpose an

upper and lower bound ∆+, ∆− are determined that over the measurement value with the probability

7

Hubert Weber, Einführung in die Wahrs heinli hkeitsre hnung und Statistik für Ingenieure, Teuber, 1992.

56

0

0.02

0.04

0.06

0.08

0.1

0.12

0 50 100 150 200 250

y

f(y

)

m = 10

m = 100

Figure 2.6: Density fun tion f(y) of the χ2distribution for two values of m (degrees of freedom). The

area under the urve evaluated up to a ertain threshold y orresponds to the probability that the χ2

distributed random variable is smaller or equal to y.

n χ2n,0.005 χ2

n,0.010 χ2n,0.050 χ2

n,0.100 χ2n,0.900 χ2

n,0.950 χ2n,0.990 χ2

n,0.995

10 2.156 2.558 3.940 4.865 15.99 18.31 23.21 25.19

100 67.33 70.07 77.93 82.36 118.5 124.3 135.8 140.2

1000 888.5 898.9 927.6 943.1 1058 1075 1107 1119

Table 2.4: Quantiles of the χ2distribution where n orresponds to the degrees of freedom.

p. It is assumed that the measurement value lies with p/2 below the lower bound and with p/2 above

the upper bound. The bounds are then found as

∆− = 10 log

(

χ2n,[(1−p)/2]

n

)

∆+ = 10 log

(

χ2n,[(1+p)/2]

n

)

(2.21)

Table 2.5 shows a few orresponding bounds, al ulated with Eq. 2.21.

n p = 0.90 p = 0.9910 −4.0 . . .+ 2.6 dB −6.6 . . .+ 4.0 dB

100 −1.1 . . .+ 0.9 dB −1.7 . . .+ 1.5 dB

1000 −0.3 . . .+ 0.3 dB −0.5 . . .+ 0.5 dB

Table 2.5: Ranges of un ertainty in the RMS al ulation of noise-like signal as a fun tion of the degrees

of freedom n for the probabilities p of 90 and 99%.

The derivation above is based on the RMS determination over a xed time frame T . It an be shown

8

that for a moving average RMS al ulation with time onstant RC the same un ertainty is obtained

for

2RC = T (2.22)

8

C. G. Wahrmann, J. T. Bro h, On the Averaging Time of RMS Measurements, B&K Te hni al Review, No. 2 (1975).

57

2.5 Measurement instruments

2.5.1 Mi rophones

Mi rophones are transdu ers that transform an a ousti al signal into an ele tri al one. For mea-

surement purposes only omnidire tional pressure sensitive ondenser mi rophones are used. However

towards high frequen ies, pressure mi rophones loose their omnidire tionality if the sound wave length

is in the same order of magnitude as the mi rophone diameter. This denes an upper frequen y limit.

High frequen y sound waves hitting the mi rophone in dire tion of the membrane normal produ e a

pressure pile-up whi h orresponds to an in rease in sensitivity. This deviation from a at frequen y

response an range up to 10 dB. Su h a mi rophone an be used without further measures only for

sound in ident dire tion parallel to the membrane or in small avities where no wave propagation takes

pla e. Consequently these mi rophones are alled pressure response types.

Mi rophones an be designed for usage under normal in ident dire tion by a ompensation of the above

mentioned ee t by appropriate frequen y dependent attenuation. These mi rophones are alled free

eld response types. They are more ommon than pressure response mi rophones. Fig. 2.7 shows the

above mentioned pressure pile-up in form of the frequen y response of a pressure response mi rophone

for dierent sound in ident dire tions.

Figure 2.7: Frequen y response of a 1/2" pressure response mi rophone for dierent sound in ident

dire tions (B&K 4166).

Some measurement instruments allow for a sele tion of the in ident dire tion dependent frequen y

orre tion by the user. So it be omes e.g. possible to measure with a free eld mi rophone in a diuse

eld with sound in iden e equally distributed over all dire tions.

The two most important properties of a measuring mi rophone are:

• dynami range (lower limit dened by self noise, upper limit given by a spe i level of distortion)

• frequen y range

Regarding these two properties no ideal mi rophone exists. The optimization of one parameter results

in a deterioration of the other. Table 2.6 shows spe i ations of typi al measuring mi rophones.

2.5.2 Calibrators

Calibrators are devi es that an be mounted on mi rophones and produ e a highly stable and repro-

du ible sound pressure. Calibrators are used prior to a measurement to alibrate the mi rophone and

the instrument. There are two ommon types:

• pistonphone

58

mi rophone diameter in (1 = 2.5 m) dynami range frequen y range

1 10 dB(A). . .146 dB(A) 2 Hz . . .18 kHz

1/2 15 dB(A). . .146 dB(A) 2 Hz . . .20 kHz

1/4 29 dB(A). . .164 dB(A) 2 Hz . . .100 kHz

Table 2.6: Spe i ations of typi al measuring mi rophones of varying diameter.

• a ousti al alibrator

The pistonphone generates the referen e sound pressure by the movement of two small pistons with

extremely pre ise lift. It operates at a frequen y of 250 Hz and produ es a nominal sound pressure

level of 124 dB (+/- 0.15 dB). As the produ ed sound pressure depends on the density of the air a

orre tion for the ambient air pressure is ne essary.

The a ousti al alibrator generates the referen e sound pressure by aid of a small loudspeaker. Usually

a frequen y of 1 kHz is used, the sound pressure level is typi ally 94 dB and possibly 114 dB with a

reprodu ibility of +/- 0.3 dB. The ex itation frequen y of 1 kHz has the advantage that it doesn't

matter if the A-lter is involved as the A-lter is transparent at 1 kHz.

2.5.3 Sound level meter

The sound level meter is the standard measuring instrument of the a ousti ian. Today's instruments

operate digitally. They measure sound pressure and allow for the evaluation of a variety of signal

attributes su h as maximum and minimum levels, equivalent energy levels and event levels. Figure 2.8

shows the blo k diagram of a sound level meter.

Figure 2.8: Blo k diagram of a sound level meter.

Fun tional units of a sound level meter :

mi rophone and amplier the mi rophones used are omnidire tional ondenser mi rophones, usually

prepolarized.

able the mi rophone able represents a signi ant load for the mi rophone apsule. To drive su h a

load, a mi rophone amplier is absolutely ne essary. Long ables an lead to nonlinear distortions

at high levels and high frequen ies.

input amplier the input amplier allows for a stepwise adaptation of the measuring range to the

signal. The dynami range of sound level meters is typi ally in the order of 80 dB.

weighting lter A- or C-weighting an be applied to a ount for the frequen y response of the human

hearing. Some instruments allow to insert external lters.

integrator dierent signal attributes are evaluated simultaneously and stored for the presentation in

the display.

display indi ation of the sele ted signal attribute.

The International Ele trote hni al Commission (IEC) has spe ied requirements for lass 1 (pre ision)

and lass 2 sound level meters

9

. Measurements in onne tion with the Swiss noise legislation

10

have to

9

IEC Norm 61672 Ele troa ousti s - Sound level meters, 2002-05.

10

LSV: S hweizeris he Lärms hutzverordnung, http://www.admin. h/ h/d/sr/ 814_41.html

59

fulll the requirements a ording to lass 1. Furthermore the instruments need approval from METAS,

the Federal O e of Metrology. All instruments have to be initially alibrated by METAS. Every two

years the instruments need a re alibration by METAS or by a ertied body.

2.5.4 Level re orders

Level re orders an register the level time history of an a ousti al signal. Today's level re orders operate

digitally. They write the information in a memory for further data pre essing and evaluation.

2.5.5 Analyzers for level statisti s

Analyzers for level statisti s allow for the evaluation of statisti al quantities su h as L1 or L50. Theyindi ate the levels that are ex eeded during 1% (L1) or 50% (L50) of the measurement time. In today's

pra ti e, statisti al levels play a minor role. One reason is the fa t that due to nonlinear behavior it is

not possible to perform al ulations based on these quantities.

2.5.6 Frequen y analyzers

With help of frequen y analyzers it is possible to investigate the spe tral ontents of a signal. In many

areas of signal pro essing FFT analyzers are ommon. For a ousti al appli ations on the other hand,

often a frequen y resolution that is onstant relative to frequen y is preferred (e.g. o tave and third-

o tave bands). For spe ial purposes even smaller bandwidths (one sixth or one twelfth of an o tave) are

available. Frequen y analyzers are apable to perform the bandpass ltering in real-time simultaneously

in a range from to 20 Hz to 20 kHz. Figure 2.9 shows a hand-held two- hannel analyzer with a maximal

temporal resolution of 5 ms.

Figure 2.9: Example of a hand-held third-o tave band analyzer.

2.5.7 Sound re orders

It is often useful to re ord the mi rophone signal with a sound re order for possible additional subsequent

analysis. Today's state of the art are portable digital re orders. They oer a frequen y range up to

at least 20 kHz and a dynami range of 90 dB or more. To establish a relation to an absolute signal

level, the alibration tone is re orded at the beginning of a measurement. A repetition at the end of

the re ording allows for a ontrol that the properties of the measurement hain haven't hanged.

60

2.6 Spe ial measurement tasks

2.6.1 Sound intensity measurements

Sound intensity meters an apture and evaluate sound intensity. Sound intensity is a ve tor quantity

and has thus an orientation. The intensity an be al ulated as produ t of sound pressure and sound

parti le velo ity (Eq. 2.23).

∣

∣

∣

~I∣

∣

∣ = p(t) |~v(t)| (2.23)

While the measurement of sound pressure is relatively simple, the sound parti le velo ity is mu h more

di ult to apture. A interesting development in this ontext is the mi roown transdu er

11

,

12

,

13

.

The prin iple behind the mi roown transdu er is a of hot-wire anemometer rea ting dire tly on the

sound parti le velo ity. The transdu er an be built with dimensions mu h smaller than the wave

lengths of interest in the audio range. However the frequen y range is limited towards high frequen-

ies by the fa t that the heating and ooling of the wires needs some time due to their thermal apa ity.

Still a ommon method to evaluate sound parti le velo ities is the two mi rophone te hnique. It uses

the relationship between the temporal derivative of the sound parti le velo ity and the lo al derivative

of the sound pressure:

ρ∂vx∂t

= − ∂p

∂x(2.24)

where vx is the sound parti le omponent in the x-dire tion. Integration yields

vx = −1

ρ

∫

∂p

∂xdt (2.25)

The partial derivative of sound pressure relative to the x omponent an be approximated by a nite

dieren e:

∂p

∂x=p(x+∆x)− p(x)

∆x(2.26)

where p(x) and p(x + ∆x) orrespond to the sound pressure at positions x and x + ∆x. The

approximation by a dieren e is valid only if ∆x is mu h smaller than the proje tion of the wave length

onto the x axis.

With Eq. 2.26 inserted in Eq. 2.25 and Eq. 2.25 in Eq. 2.23 the x- omponent of the intensity nally

be omes - expressed in p(x) and p(x+∆x)

Ix =1

T

T∫

0

(

−1

2

1

ρ∆x(p(x) + p(x+∆x))

∫

p(x)− p(x+∆x)dt

)

dτ (2.27)

where

T : time of integration (averaging)

The availability of sound intensity allows for an elegant measurement of the sound power of a

sour e

14

. To do so the sound sour e is surrounded by a losed virtual surfa e. At representa-

tive points on this surfa e the normal omponent of sound intensity is measured. By multipli ation

with the orresponding areas and summation the total emitted sound power of the sour e is determined.

11

Jörg Sennheiser, MICRO-MINIATURIZED MICROPHONE FOR COMMUNICATION APPLICATIONS, 2nd Conven-

tion of the EAA, Berlin, 1999.

12

W.F. Druyvesteyn, H.E. de Bree, A Novel Sound Intensity Probe Comparison with the Pair of Pressure Mi rophones

Intensity, Journal of the Audio Engineering So iety, vol. 48, p.49-56 (2000).

13

R. Raangs et al., A Low-Cost Intensity Probe, Journal of the Audio Engineering So iety, vol. 51, p.344-357 (2003).

14

ISO Norm 9614-1,2 A ousti s - Determination of sound power levels of noise sour es using sound intensity; Measure-

ment at dis rete points and measurement by s anning. 1993, 1996.

61

Additional sound power measurement strategies

If no intensity measurement is available, the sound power of a sour e an be estimated by pure sound

pressure measurements alone:

A rst method is to install the sour e in a reverberant room and to measure the sound pressure in the

diuse eld. From this sound pressure and with knowledge of the sound absorption in the reverberant

room, the sound power an be determined

15

,

16

.

In the se ond arrangement the sour e is installed in an absorbing environment above a ree ting

ground. This an be in an ane hoi room or outdoors. At several positions in dened distan e from

the sour e the sound pressure is evaluated. If the distan e from the mi rophones to the sour e is

large enough so that near eld ee ts an be negle ted, the sound parti le velo ity an be dedu ed

from sound pressure. The sound power of the sour e is then evaluated analogously to the ase where

intensity is measured dire tly

17

,

18

,

19

,

20

.

The third method is based on the omparison of the sour e under onsideration with a referen e sour e

of known sound power. The sound pressures produ ed by the two sour es are measured in the diuse

eld of an environment with not too mu h absorption. The ratio of the square of the two sound pressure

values orresponds to the ratio of the sound power of the two sour es

21

.

2.6.2 System identi ation

General

A ommon task in the eld of a ousti s is the des ription of the transmission properties of systems with

an input x and an output y. In many systems the input and output are dierent physi al quantities, as

e.g. in ase of a loudspeaker with an ele tri al input and an a ousti al output. Here it is assumed that

the systems are linear and time invariant whi h means that they don't hange their properties over time.

There are two fundamental possibilities for the des ription of su h a system. In the time domain it is

the impulse response h(t), in the frequen y domain the frequen y response H(ω). Both representationsdes ribe the system ompletely. By help of the Fourier transformation they an be onverted one into

the other.

H(ω) =

+∞∫

−∞

h(t)e−jωtdt (2.28)

h(t) =1

2π

+∞∫

−∞

H(ω)ejωtdω (2.29)

In the time domain the output y(t) of the system is given as the onvolution of the input x(t) with the

impulse response h(t):

y(t) =

t∫

−∞

x(τ)h(t − τ)dτ (2.30)

15

ISO Norm 3741 A ousti s - Determination of sound power levels of noise sour es using sound pressure. Pre ision

methods for reverberation rooms. Draft 1998.

16

ISO Norm 3742 A ousti s - Determination of sound power levels of noise sour es. Pre ision methods for dis rete-

frequen y and narrow-band sour es in reverberation rooms. 1988.

17

ISO Norm 3744 A ousti s - Determination of sound power levels of noise sour es using sound pressure. Engineering

method in an essentially free eld over a ree ting plane. 1994.

18

ISO Norm 3745 A ousti s - Determination of sound power levels of noise sour es. Pre ision methods for ane hoi

and semi-ane hoi rooms. 1977.

19

ISO Norm 3746 A ousti s - Determination of sound power levels of noise sour es using sound pressure. Survey

method using an enveloping measurement surfa e over a ree ting plane. 1995.

20

ISO Norm 3748 A ousti s - Determination of sound power levels of noise sour es. Engineering method for small,

nearly omnidire tional sour es under free-eld onditions over a ree ting plane. 1983.

21

ISO Norm 3747 Akustik - Bestimmung der S hallleistungpegel von Geräus hquellen aus S halldru kmessungen -

Verglei hsverfahren zur Verwendung unter Einsatzbedingungen. Entwurf 1998.

62

In the frequen y domain the system output Y (ω) orresponds to the produ t of the input X(ω) andthe frequen y response H(ω):

Y (ω) = X(ω)H(ω) (2.31)

A serious di ulty with the pra ti al measurement of the properties of a system is the presen e of

unwanted noise. Usually the noise adds at the output of the system as depi ted in Fig. 2.10. Only the

signal v(t) an be measured as superposition of the system output y(t) and the noise n(t).

The question arises, in whi h way a statement about the system output y(t) is possible. If it is

su ient to determine the signal power of y, the noise power n2rms an be estimated (with input

x(t) = 0) and subtra ted from v2rms, yielding an estimate for y2rms. This works under the assumption

that no orrelation exists between the unwanted noise n(t) and the system input x(t). Furthermore it

is ne essary that n(t) is stationary, meaning it doesn't hange its properties over time.

Figure 2.10: Identi ation of the system h with additional superposition of unwanted noise at the

output.

A more sophisti ated approa h is the usage of orrelation fun tions. Hereby auto orrelation fun tions

Rxx(τ) and ross orrelation fun tions Rxy(τ) are needed a ording to the following denitions:

Rxx(τ) =1

2T

+T∫

−T

x(t − τ)x(t)dt (T → ∞) (2.32)

Rxy(τ) =1

2T

+T∫

−T

x(t − τ)y(t)dt (T → ∞) (2.33)

The ross orrelation between the input and output of a linear system with impulse response h(t) anbe written as (with T → ∞):

Rxy(τ) =1

2T

+T∫

−T

x(t − τ)

∞∫

0

x(t− u)h(u)dudt

=

∞∫

0

h(u)1

2T

+T∫

−T

x(t− τ)x(t − u)dtdu

=

∞∫

0

h(u)1

2T

+T∫

−T

x(t − (τ − u))x(t)dtdu

=

∞∫

0

h(u)Rxx(τ − u)du

= h(t) ∗Rxx(τ) (2.34)

where

∗: onvolution

The relation (2.34) is alledWiener-Hopf equation. With known auto orrelation fun tion of the stimulus

x(t) the system impulse response an be determined from the measured ross orrelation fun tion

63

between the input and output. The big advantage of evaluation of the ross orrelation fun tion is the

fa t that un orrelated noise an els out perfe tly in the limiting ase of innite measuring time. Applied

to the system identi ation task from Fig. 2.10 it an be on luded that Rxy = Rxv and therefore

Rxx(τ) ∗ h(t) = Rxv(τ) (2.35)

In the limiting ase of an innitely long measurement, the impulse response h(t) an thus be determined

perfe tly with help of Eq. 2.35.

The Wiener-Khin hine theorem states that the auto orrelation fun tion Rxx(τ) and the power spe trumGxx(ω) as well as the ross orrelation fun tionRxy(τ) and the ross power spe trumGxy(ω) are relatedby the Fourier transformation. Consequently Eq. 2.34 an be translated into the frequen y domain as

H(ω) =Gxy(ω)

Gxx(ω)(2.36)

where

Gxx(ω) : power spe trum of the input signal x(t)Gxy(ω) : ross power spe trum of the input signal x(t) and output signal y(t)

The power spe trum Gxx(ω) and the ross power spe trum Gxy(ω) are given as

Gxx(ω) = E[X∗(ω)X(ω)] (2.37)

Gxy(ω) = E[X∗(ω)Y (ω)] (2.38)

where

E : expe tation value

X(ω) : Fourier transform of the input signal x(t)Y (ω) : Fourier transform of the output signal y(t)∗: omplex onjugate

In the ontext of system identi ation, the oheren e γ2xy(ω) fun tion is often evaluated to des ribe

the quality of the measurement. The oheren e is dened as

γ2xy(ω) =

∣

∣Gxy(ω)2∣

∣

Gxx(ω)Gyy(ω)(2.39)

If there is a stri t linear relationship between the input and output of a system, the oheren e γ2xy(ω)equals 1 everywhere. If there is no orrelation at all between the input and output the oheren e

be omes 0. In pra ti al appli ations the oheren e is usually a little below 1, meaning that

• the measurement is distorted by noise and/or

• input and output are related not only linearly and/or

• the output depends on the input but is inuen ed by further quantities

Correlation measurement in the time domain

The orrelation measurement in the time domain evaluates the impulse response of a system by evalu-

ation of the ross orrelation fun tion between the input and output signal a ording to Eq. 2.34. If a

stimulus with a dira -like auto orrelation fun tion is hosen, Eq. 2.34 simplies to

h(t) = Rxy(τ) (2.40)

where

h(t): impulse response of the system

Rxy(τ): ross orrelation fun tion between input and output

Dierent stimulus signals with dira -like auto orrelation fun tions are worth to be onsidered. White

noise for example is one possibility. An other interesting signal lass are two-valued pseudo random

sequen es or maximum length sequen es (MLS) . Su h sequen es s(k) an be found for lengths L with

64

L = 2n − 1 n: integer > 0 (2.41)

-2

-1

0

1

2

Figure 2.11: Part of a two valued pseudo random sequen e. The sequen e values 0 are mapped to +1,

sequen e values 1 are mapped to -1.

Pseudo random sequen es an be generated with help of shift registers. The se ret lies in a suitable

ex lusive-or operation and feed-ba k of the orre t digits. Table 2.7 shows for dierent orders nexamples of primitive polynomials. Fig. 2.12 shows exemplarily the translation of a primitive polynomial

into a feed-ba k stru ture of the shift register. More details about primitive polynomials an be found

in the book by Weldon

22

.

order n primitive polynomial order n primitive polynomial

1 x+ 1 16 x16 + x5 + x3 + x2 + 12 x2 + x+ 1 17 x17 + x3 + 13 x3 + x+ 1 18 x18 + x7 + 14 x4 + x+ 1 19 x19 + x6 + x5 + x+ 15 x5 + x2 + 1 20 x20 + x3 + 16 x6 + x+ 1 21 x21 + x2 + 17 x7 + x+ 1 22 x22 + x+ 18 x8 + x6 + x5 + x+ 1 23 x23 + x5 + 19 x9 + x4 + 1 24 x24 + x4 + x3 + x+ 110 x10 + x3 + 1 25 x25 + x3 + 111 x11 + x2 + 1 26 x26 + x8 + x7 + x+ 112 x12 + x7 + x4 + x3 + 1 27 x27 + x8 + x7 + x+ 113 x13 + x4 + x3 + x+ 1 28 x28 + x3 + 114 x14 + x12 + x11 + x+ 1 29 x29 + x2 + 115 x15 + x+ 1 30 x30 + x16 + x15 + x+ 1

Table 2.7: Examples of primitive polynomials of order n 1. . .30.

Ausgang1xx

2x

3x

4x

5x

6x

7x

8

Figure 2.12: Feed-ba k stru ture for the primitive polynomial x8 + x6 + x5 + x+ 1.

If maximum length sequen es s(k) are repeated periodi ally, the auto orrelation fun tion Rss(k) be- omes:

Rss(k) =

1 : k = iL, i : integer ≥ 0−1/L : else

(2.42)

For large L the fun tion Rss is a good approximation of the Dira pulse. Indeed this holds only within

one period of length L. It has to be assured that the system impulse response drops o to small enough

22

Error-Corre ting Codes, W. Wesley Peterson, E. J. Weldon, MIT-Press 1972.

65

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 20 40 60 80 100

Figure 2.13: Auto orrelation fun tion Rss(k) of a periodi ally repeated maximum length sequen e of

length L = 15.

values to avoid temporal aliasing. Due to their periodi ity the spe trum of maximum length sequen es

is a line spe trum. The separation between two frequen y lines ∆f is

∆f =1

L∆t(2.43)

where

L: sequen e length∆t: 1/ lo k frequen y

The envelope E(f) is given by

E(f) =sin2

(

πffc

)

(

πffc

)(2.44)

where

fc: lo k frequen y

Up to about half of the lo k frequen y, the spe trum of a maximum length sequen e is at. A big

advantage ompared to white noise is the signi antly lower rest fa tor

23

.

Compared to single impulse measurements, the orrelation te hnique shows a tremendously improved

signal-to-noise ratio. The orrelation pro ess a tually performs an average over L single impulse mea-

surements where L is the sequen e length. During the averaging, the signal of interest adds linearly

with orre t phase while the noise adds on a square basis only. Thus for ea h doubling of the sequen e

length the signal/noise ratio improves by 3 dB. Relative to a single impulse the signal-to-noise ratio

improvement G an be written as:

G = 3 log2(L) [dB] (2.45)

In pra ti al appli ations, sequen e lengths in the order of 100'000 are used, whi h orresponds to

a S/N improvement of about 50 dB. Impulse response measurements based on maximum length

sequen es MLS are widely used in room a ousti s

24

,

25

,

26

.

MLS measurements may also be interesting in situations where the fo us lies not on the exa t ourse of

the impulse response but on the total energy that is transferred by a system. This energy an be found

by integration of the squared impulse response. An interesting property of the MLS te hnique is the fa t

that any disturbing noise during the measurement is mapped onto a stationary noise-like signal that is

23

The rest fa tor des ribes the ratio of the peak value to the root mean square value of the signal.

24

Kenneth W. Go , Appli ation of orrelation te hniques to some a ousti measurements, Journal of the A ousti al

So iety of Ameri a, 1955, v.27, p.236.

25

M. R. S hroeder, Integrated impulse method measuring sound de ay without using impulses, Journal of the A ousti al

So iety of Ameri a,1979, v.66, p.497-500.

26

R. Bütikofer, K. Bas hnagel, Bauakustis he Messungen mit MLS; Konsequenzen für das Bes hallungssystem,

Forts hritte der Akustik, DAGA 98 (1998), p.652-653.

66

equally smeared over the impulse response. The power of this unwanted noise an be estimated in a re-

gion of the impulse where no signal is present (e.g. during the time until the arrival of the dire t sound).

The appli ation of maximum length sequen es as stimulus allows to use the Hadamard transformation

for a very e ient al ulation of the ross orrelation fun tion

27

.

As already mentioned, the resulting system impulse response is periodi with a period length that

orresponds to the stimulus sequen e length. In order to avoid temporal aliasing (overlapping) it has

to be assured that this sequen e length is larger than the length of the impulse response. In room

a ousti al appli ations the impulse response length an be assumed as the reverberation time.

The usage of the orrelation measurement te hnique is only possible, if the system under investigation

is linear and time invariant. If these onditions are not fullled an additional noise omponent o urs in

the resulting impulse response

28

,

29

,

30

. Typi al ases where MLS doesn't work are measurements with

moving loudspeakers and/or mi rophones. Strong turbulent air ows are problemati as well, limiting

the appli ability outdoors. A serious sour e of non-linearity are loudspeakers that are driven with high

amplitudes. For non-linear systems, measurements with sweeps as stimulus are favorable

31

.

Time - bandwidth un ertainty prin iple

In many ases one is interested in an bandpass ltered impulse response. However, any ltering produ es

a temporal smearing. The time-bandwidth un ertainty prin iple states that the produ t of temporal

un ertainty and analysis bandwidth an not drop below a ertain limit. The more narrow the analysis

bandwidth, the larger is the temporal un ertainty whi h an be des ribed by a minimal pulse width

32

.

If the temporal un ertainty is dened as the -3 dB width of the bandlimited impulse, the un ertainty

prin iple says

∆t∆f ≥ 0.5 (2.46)

with

∆t: temporal width of the impulse in se onds, evaluated at the -3 dB points

∆f : frequen y bandwidth in Hz

The interesting question is, what kind of bandpass lter fun tion of given width produ es a minimal

pulse width enlargement so that the equal sign holds in Eq. 2.46. A bri k wall band pass lter, for

example, leads to a ∆t∆f produ t of 1. This is a fa tor 2 away from the optimum. It an be shown

that the optimal band lter has a frequen y response a ording to Eq. 2.47.

G(ω) = 0.5

√π

α

(

e−(ω+ω0)2/4α2

+ e−(ω−ω0)2/4α2

)

(2.47)

where

ω0: enter frequen y of the bandpass lter in rad/s

α = ∆ω√2π

∆ω: lter bandwidth in rad/s

In the time domain the frequen y response of Eq. (2.47) orresponds to the so alled Gabor pulse

33

:

g(t) = e−α2t2 cos(ω0t) (2.48)

27

J. Borish, J. B. Angell, An e ient algorithm for measuring the impulse response using pseudo random noise, Journal

of the Audio Engineering So iety, 1983, v.31, p.478-487.

28

J. Vanderkooy, Aspe ts of MLS Measuring Systems, Journal of the Audio Engineering So iety, vol. 42, p.219-231

(1994).

29

C. Dunn, M. O. Hawksford, Distortion Immunity of MLS-Derived Impulse Response Measurements, Journal of the

Audio Engineering So iety, vol. 41, p.314-335 (1993).

30

U. P. Svensson, J. L. Nielsen, Errors in MLS Measurements Caused by Time Varian e in A ousti Systems, Journal

of the Audio Engineering So iety, vol. 47, p.907-927 (1999).

31

G. Stan, J. Embre hts, D. Ar hambeau Comparison of Dierent Impulse Response Measurement Te hniques Journal

of the Audio Engineering So iety, vol. 50, no. 4, p.249-262 (2002).

32

J. S. Suh, P. A. Nelson, Measurement of transient response of rooms and omparison with geometri al a ousti

models, J. A ousti al So iety of Ameri a, vol. 105, p. 2304-2317 (1999).

33

D. Gabor, Theory of Communi ation, J. IEEE, London, vol. 93(III), p.429-457 (1946).

67

The width of the Gabor pulse (2.48) evaluated at the -3 dB points is found to:

∆t =

√

π

2

1

α(2.49)

Figure 2.14 shows an example of su h an optimal bandpass lter frequen y response and the orre-

sponding time response.

0 200 400 600 800 1000

frequency [Hz]

0

0.2

0.4

0.6

0.8

1

filt

er

gain

−0.01 −0.005 0 0.005 0.01

time [s]"

−1

−0.5

0

0.5

1

am

plitu

de

Figure 2.14: Filter frequen y response of an optimal bandpass lter a ording to Eq. 2.47 and the

orresponding time response of a ltered impulse (2.48) for a enter frequen y of 500 Hz (ω0 = 3142

rad/s) and a bandwidth of 300 Hz (∆ω = 1885 rad/s).

2.6.3 Measurement of reverberation times

Introdu tion

The reverberation time is an important quantity to des ribe the a ousti al property of rooms.

Reverberation stands for the delayed rea tion of a room to temporally varying ex itation. If a sour e

of onstant level is swit hed on, the sound travels as dire t sound to a re eiver position, followed by

ree tions with in reasing temporal density. A few tenths of a se ond after the swit h-on moment, a

stationary ondition is a omplished with onstant sound energy density in the room. This ondition

represents the equilibrium state where the sound power fed by the sour e equals the sound power that

is absorbed in the air and at the boundary.

The reverberation pro ess itself manifests after swit hing o the sour e. After the traveling time from

the sour e to the re eiver, the ontribution of the dire t sound disappears. The great number of

ree tions however still make their way to the re eiver. With time these ree tions be ome weaker

and weaker due to absorption in the air and at the boundaries. The sound pressure drops more or

less exponentially, whi h means that the sound pressure level follows a straight line. The time span

measured from the moment when the sour e is swit hed o until the level drops for 60 dB is alled

reverberation time T 60. Typi al values for reverberation times lie between a few tenth of a se ond

(living rooms) and several se onds (large hur hes).

As the absorption properties of the room boundaries are frequen y dependent, the reverberation times

are frequen y dependent. Consequently the reverberation times are evaluated in third o tave or o tave

bands.

S hroeder reverse integration

The lassi al method to determine reverberation times is to use a loudspeaker that emits a random

signal whi h is swit hed o after a ertain time. The observation of the sound pressure level as a

fun tion of time will show random variations that dier from measurement to measurement. The

reason for this is the random phase of the room modes at the swit h-o moment (Fig. 2.15).

To eliminate these variations and to smoothen the level - time urves, the measurements have to be

performed several times and averaged. S hroeder

34

has shown that the average of n measurements

with n → ∞ an be found by one measurement alone. To do so one has to determine the squared

34

M. R. S hroeder, New Method of Measuring Reverberation Time, Journal of the A ousti al So iety of Ameri a, 1965,

p.409-412.

68

−1 −0.5 0 0.5 1 1.5

time [s]

−40

−30

−20

−10

0

10

sound p

ressure

level [d

B]

−1 −0.5 0 0.5 1 1.5

time [s]

−40

−30

−20

−10

0

10

sound p

ressure

level [d

B]

−1 −0.5 0 0.5 1 1.5

time [s]

−40

−30

−20

−10

0

10

sound p

ressure

level [d

B]

Figure 2.15: Classi al measurement of the de ay in a room after swit hing o the sour e (top and

middle). The bottom urve is found with the S hroeder reverse integration of the squared impulse

response.

impulse response r2(t) of the room for the sour e and re eiver position under onsideration. By a so

alled reverse integration it is then found how the squared sound pressure dies away on average (2.50).

〈s2(t)〉 ∼∞∫

t

r2(τ)dτ (2.50)

where

〈s2(t)〉: average of all possible de ays of the squared time response

r2(t): squared impulse response of the room for the sele ted sour e and mi rophone positions

69

Measurement of short reverberation times at small lter bandwidths

Often the reverberation is measured in third o tave bands. At the lowest third o tave bands the

bandwidth is so small that the lter applied to the impulse response may dominate the de ay pro ess

(Fig. 2.16).

0 0.05 0.1 0.15 0.2 0.25

time [s]

amplitude

Figure 2.16: Impulse response of a third o tave band lter at 63 Hz.

As a rule of thumb it an be on luded that the following ondition has to be fullled in order to

guarantee a valid reverberation time measurement

35

:

B × T 60 > 16 (2.51)

where

B: bandwidth of the lter

T 60: reverberation time.

The impulse response of a bandpass lter is asymmetri al (Fig. 2.16). It is therefore bene ial to

reverse the time axis

36

,

37

,

38

. This an be done either by playing a re orded signal ba kwards or by

using a lter with time reversed impulse response. In both ases the frequen y ontent remains the

same. Compared to the ondition in Eq. 2.51, a fa tor 4 an be gained, meaning that only the following

ondition has to be fullled:

B × T 60 > 4 (2.52)

where

B: bandwidth of the lter

T 60: reverberation time.

Fig. 2.17 shows the signi antly steeper de ay of the reverse integrated impulse response of the time

reversed lter ompared to the normal lter.

2.7 Pressure zone mi rophone onguration

Often an a ousti al measurement should provide information about the dire t sound or the sound

power of a sour e. In these ases the sound ree tion at the ground is parti ularly disturbing, as

interferen e o urs in ombination with the dire t sound. If the ground surfa e is a ousti ally hard

35

F. Ja obsen, A note on a ousti de ay measurements, Journal of Sound and Vibration, v.115, 1987.

36

F. Ja obsen, J. H. Rindel, Time reversed de ay measurements, Journal of Sound and Vibraiton, v.117, p.187-190,

1987.

37

B. Rasmussen, J. H. Rindel, H. Henriksen, Design and Measurement of Short Reverberation Times at Low Frequen ies

in Talk Studios, Journal of the Audio Engineering So iety, v. 39, n. 1/2, p.47-57, 1991.

38

M. A. Sobreira-Seoane, D. Pérez Cabo, F. Ja obsen, The inuen e of the group delay of digital lters on a ousti

de ay measurements, Applied A ousti s, v. 73, p. 877-883, 2012.

70

−0.2 −0.1 0 0.1 0.2

time [s]

−30

−20

−10

0

level [d

B]

normal filter

reversed filter

Figure 2.17: De ay (S hroeder reverse integration) of the normal and the time reversed third o tave

lter at 63 Hz.

it is possible to put the omnidire tional, pressure sensitive mi rophone dire tly on the ground. This

set-up is alled pressure zone onguration. Independently of the angle of in iden e the sound

pressure of the in iden e wave doubles on the hard surfa e. In the dB s ale this orresponds to

a 6 dB in rease relative to the dire t sound in the free eld. A prerequisite is that the ree ting

surfa e is large enough. The ondition large enough an not easily be onverted into spe i dimensions.

Here a measurement is shown for a ree ting plate of 1.50×1.40 m. In the enter of the plate a

1/2 mi rophone was installed with the membrane parallel to the plate surfa e in a distan e of 2 mm.

A loudspeaker in a distan e of 2.70 m was used as sour e and emitted pink noise. The angle of

in iden e φ relative to the plate normal dire tion was varied between 0 and 90

. As the mi rophone

pointed to the plate, the 0

dire tion orresponded to an angle of 180

for the mi rophone. Figure

2.18 shows the measured third o tave band sound pressure levels relative to free eld as a fun tion of

φ. In the mid-frequen y range for not grazing in iden e the onguration produ es the expe ted 6 dB

ampli ation. For low frequen ies and/or grazing in iden e the ampli ation is signi antly redu ed

due to insu ient size of the ree ting surfa e. At higher frequen ies the ampli ation drops due to

the de reasing sensitivity of the mi rophone itself for o-axis in ident.

100 160 250 400 630 1k 1k6 2k5 4k 6k3 10k 16k

third octave band [Hz]

−4.0

−2.0

0.0

2.0

4.0

6.0

8.0

10.0

sound p

ressure

level re

. fr

ee f

ield

[dB

]

80°

60°

40°

20°

0°

Figure 2.18: Third o tave band levels relative to free eld for a 1/2 mi rophone in pressure zone

onguration on a plate with dimensions 1.50×1.40 m for dierent angles of in iden e.

71

2.8 Un ertainty of a ousti al measurements

In almost all ases a ousti al measurements in lude unwanted ee ts. If the un ertainty due to these

ee ts is too large, the results may be ome worthless. For the ase of determining the sound pressure

at a ertain lo ation, the following aspe ts have to be onsidered:

• the sour e may be in a not representative ondition

• the propagation medium may be in a not representative ondition (e.g. upwind onditions and a

negative verti al temperature gradient)

• the surrounding of the mi rophone may inuen e the measurement in a non representative way

• the un ertainty of the alibration and toleran es of the measurement instrument

• possible unwanted disturbing noise (often this is the main di ulty and appropriate strategies

have to be found to remove or ex lude this noise. If this noise is un orrelated and stationary, its

ontribution an be estimated and subtra ted on a power basis)

• Un ertainty in the determination of the power of random signals

For ea h measurement the total un ertainty has to be spe ied. Typi al values are in the range of

±1..3 dB in the sense of a standard deviation.

72

Chapter 3

The human hearing

3.1 Stru ture and prin iple of operation of the ear

TympanicCavity

ossicles

AuditoryNerve

Eustachian TubeTympanicMembrane

ExternalAuditory Canal

Stapes(attached to oval window)

Cochlea

RoundWindow

Figure 3.1: Se tion through the human ear. sour e: Chittka L, Bro kmann

The human ear an be separated into three main parts, the outer ear, the middle ear that is lled with

air and the inner ear or o hlea, lled with a uid. The outer ear omprises the auri le and the outer

ear anal. It is separated from the middle ear by the tympani membrane or ear drum. The middle ear

is usually losed airtight. However the Eusta hian tube provides a onne tion to the throat and allows

for pressure equalization. This an be provoked by swallowing. This onguration with a membrane

on top of a losed avity - and thus exposed to the sound eld on one side only - orresponds to a

sound pressure re eiver.

The vibrations of the tympani membrane are transmitted to the inner ear by tiny bones (ossi les).

These bones onvert the relative large ex ursions of the tympani membrane into small ex ursions at

the input of the inner ear. The benet of this transformation is an ampli ation of the for e whi h is

ne essary to ex ite the uid. The onguration performs an impedan e adjustment between air and

uid.

The ossi les in the middle ear are onne ted to mus les that an inuen e the transmission prop-

erties. If very loud sound signals are per eived, these mus les are ontra ted by reex and lower

the sensitivity of the ear and thus provide a ertain prote tion of the inner ear. The inner ear

is formed by the o hlea. The o hlea is separated into two hannels by the basilar membrane.

At the far end these two hannels are onne ted to ea h other. The uid in the two hannels

in the inner ear is ex ited by me hani al vibration of a membrane that is put into motion by

73

the ossi les. As a onsequen e of this ex itation a traveling wave is formed that runs along the

basilar membrane. The amplitudes of the traveling waves are very small. For stimuli that are

just audible they are in the order of the diameter of an atom. The lo ation of highest amplitude

depends on the frequen y of the stimulus. Thus in the inner ear a transformation takes pla e that

maps frequen y to lo ation. This me hanism is fundamental for the frequen y dis rimination of the ear.

The lo ation on the basilar membrane for maximal amplitude an be des ribed by Eq. 3.1

1

.

f = 165.4(100.06x − 1)

x =1

0.06log

(

f + 165.4

165.4

)

(3.1)

where

f : frequen y in [Hz

x: position of maximum ex ursion of the basilar membrane in [mm

The movement of the basilar membrane is dete ted by hair ells that sit on top of the membrane.

The stimulated hair ells emit ele tri al impulses that are transported to the brain by the auditory

nerve. The ear has an ex ellent frequen y dis rimination whi h an not be explained by the frequen y

dependent amplitudes of the traveling waves alone. Re ent investigations have demonstrated that

feed-ba k ee ts play an important role. There is experimental eviden e that the outer hair ells are

put into motion a tively and by this inuen e the movement of the basilar membrane. This a tivity

leads on her part to an ex itation of the ossi les and the tympani membrane and an be dete ted

by a mi rophone in the ear anal. This phenomenon is alled otoa ousti emission. Sometimes these

emissions o ur spontaneously. More relevant is the fa t that su h an emission results always as a

rea tion of the ear to an a ousti al stimulus, however only if the ear fun tions properly. These tests

are performed most easily with a short li k as stimulus. The rea tion of the ear an then be dete ted

with a delay of a few millise onds. This is an ex ellent possibility to investigate the proper working of

the ear in an obje tive manner without the need of a response of the human being. Many hospitals

use this method to dete t possible malfun tioning of the auditory system of newborns.

An ex ellent overview of physiologi al and psy hologi al aspe ts of the human ear an be found in the

book by Fastl and Zwi ker

2

3.2 Properties of the auditory system for stationary signals

3.2.1 Loudness

The intensity of the sensation of a sound is des ribed by its loudness. The are two s ales in use.

Loudness an be expressed on a linear s ale, alled sone, or on a logarithmi s ale as loudness level

LN in phon.

The loudness of a spe i sound is investigated by subje tive omparison with a referen e signal,

usually a 1 kHz tone or 1 kHz narrow band noise. The referen e signal is adjusted in su h a way that

the two sounds are per eived as equally loud. The sound pressure level in dB of the referen e signal

orresponds then to the phon value of the signal under investigation. Figure 3.2 shows urves of equal

loudness for pure tones

3

.

A urve of spe ial interest in Figure 3.2 is the auditory threshold. The urve denotes for a given

frequen y the sound pressure that is ne essary to make the tone just audible. The standard ISO 389-7

4

des ribes the threshold of hearing for binaural hearing of pure tones under free eld onditions. The

polynomial approximation in Eq. 3.2 reprodu es the tabulated values for frequen ies between 20 and

16000 Hz with an a ura y better than 0.5 dB.

1

D. D. Greenwood, A Co hlear Frequen y-Position Fun tion for Several Spe ies - 29 Years Later, J. of the A ousti al

So iety of Ameri a, vol. 87, p. 2592-2605 (1990).

2

Hugo Fastl, Eberhard Zwi ker, Psy ho-A ousti s, Springer, 2007.

3

ISO Norm 226: A ousti s - Normal equal-loudness level ontours. Se ond edition 2003.

4

ISO Norm 389-7, A ousti s - Referen e zero for the alibration of audiometri equipment - Part 7: Referen e threshold

of hearing under free-eld and diuse-eld listening onditions (1996).

74

Figure 3.2: Curves of equal loudness, labeled in phon. All ombinations of frequen y and sound pressure

level that lie on one urve result in equal loudness sensations.

T (f) ≈

2.262× 105f−3 − 3.035× 104f−2 + 2.357× 103f−1++8.3− 2.912× 10−2f + 2.2066× 10−5f2

: 20 < f ≤ 660−1.7 + 1.18247× 10−2f − 1.0653× 10−5f2 + 2.98811× 10−9f3−

−3.5279× 10−13f4 + 1.86485× 10−17f5 − 3.6299× 10−22f6: 660 < f < 16000

(3.2)

where

T (f): sound pressure level of a pure tone of frequen y f that makes the tone for binaural hearing and

under free eld onditions just audible. With in reasing age the hearing apabilities usually de rease

and thus the threshold of hearing in reases

5

.

The phon s ale orresponds to a dB s ale and is thus not proportional to the sensation. The sone s ale

on the other hand des ribes dire tly the sensation. Ea h doubling of the sone value orresponds to a

doubling of the loudness sensation. For levels not too low there is a simple onversion between phon

and sone gures. Ea h doubling of the sone value orresponds to an in rease of 10 phon. With the

denition of 1 sone ≡ 40 phon as a point of referen e the onversion an be written as:

N = 2LN−40

10(3.3)

LN ≈ 40 + 33 log(N) (3.4)

For loudness values below 40 phon, the relation from above is no longer valid. A bise tion of the sone

value is found for a phon step smaller than 10 phon.

5

ISO Norm 7029: A ousti s: Statisti al distribution of hearing thresholds as a fun tion of age (2000).

75

3.2.2 Frequen y dis rimination

The human hearing an distinguish a little more than 600 frequen y steps. For frequen ies below 500

Hz the just audible frequen y dieren e ∆f is about 3.5 Hz. Above 500 Hz the ne essary dieren e

in reases as

∆f = 0.007f (3.5)

3.2.3 Criti al bands

As mentioned above the stimulation of the ear leads to a traveling wave in the inner ear with the

onsequen e of a lo al ex itation of hair ells on the basilar membrane. Even in ase of a pure tone

stimulation, the region of ex itation has a ertain width. If the stimulus onsists of two tones of

frequen ies f1 and f2, three dierent me hanisms of per eption an be distinguished. If the dieren e

f2 − f1 is below 10 Hz, the beat an heard. If the dieren e is in reased above 10 Hz, the amplitude

modulations are no longer audible, however the beat is per eived as roughness of the sound. For further

in reasing of the frequen y dieren e this roughness disappears more and more. This point is rea hed

if both regions of ex itation on the basilar membrane do no longer overlap. This frequen y dieren e

is alled riti al band. The with of a riti al band is almost onstant below 500 Hz and amounts to

about 100 Hz. Above 500 Hz the with of the riti al bands orresponds to about 20% of the signal

frequen y. This is very lose to the bandwidth of third o tave band lters. A more a urate des ription

of the width ∆fcrit of the riti al bands is found in Eq. 3.6

6

:

∆fcrit[Hz] ≈ 25 + 75(

1 + 1.4(fS[kHz])2)0.69

(3.6)

where

fs: signal frequen y [Hz

3.2.4 Audibility of level dieren es

In a dire t A/B omparison the smallest level dieren es that are just audible are in the order of 1 dB.

Table 3.1 shows typi al level variations and the orresponding dieren es in sensation. If the two signals

are presented with a ertain time span in between, the audible dieren es are signi antly higher.

level variation sensation

0. . .1 dB not audible

2. . .4 dB just audible

5. . .10 dB learly audible

> 10 dB very onvin ing

Table 3.1: Sensation of level dieren es in a dire t A/B omparison.

3.2.5 Masking

As a onsequen e of stimulation by a tone, the basilar membrane is a tivated in a ertain region. A

se ond tone played simultaneously will only be audible if the orresponding a tivation surmounts the

one of the rst tone. In other words, the presen e of a tone leads to an upwards shift of the auditory

threshold. This shift is more pronoun ed for frequen ies higher than the tone frequen y. This shift of

the auditory threshold due to the presen e of a tone is alled masking, the tone responsible for the

masking ee t is alled masker.

3.2.6 Loudness summation

The auditory sensation in ase of the superposition of two signals distinguishes between two summation

me hanisms:

6

E. Zwi ker, R. Feldkeller, Das Ohr als Na hri htenempfänger, Hirzel Verlag, 1967.

76

In the rst ase the two signals over the same riti al band. Here the intensities add up a ording to

Eq. 3.7).

LNtotal = 10 log(

100.1LN1 + 100.1LN2)

[Phon (3.7)

where

LN1: phon gure of the rst omponent

LN2: phon gure of the se ond omponent

If for example the two signals have a strength of 50 phon ea h, the superposition yields a loudness

level of 53 phon.

In the se ond ase the two signals are learly separated in frequen y, whi h means they lie in dierent

riti al bands. Here the loudness of the signals adds up. For the example from above the two loudness

levels are onverted into the orresponding sone gures (50 phon → 2 sone). The sum of the sone

values equals 4 sone whi h in turn orresponds to 60 phon.

3.2.7 Virtual pit h

Complex tonal sounds onsist of a series of sinusoidal signals a ording to Eq. 3.8).

s(t) =

∞∑

i=1

Ai sin(iωt) (3.8)

where ω represents the angular frequen y of the fundamental, the omponents iω are the harmoni s.

The fundamental is responsible for the pit h, the harmoni s onstitute the tone olor. It happens that

the fundamental is only weak or totally missing. An example are string instruments playing tones at

low frequen ies. Nonetheless the ear an per eive the pit h of the fundamental. The auditory system

omplements the missing fundamental from the pattern of the harmoni s. This phenomenon is alled

virtual pit h. A spe iality of this virtual fundamental omponent is that it an not be masked by other

sounds.

Virtual pit h is responsible for the fa t that small loudspeakers appear to radiate sound even at low

frequen ies although this is not possible for physi al reasons. It has been proposed to manipulate audio

signals spe i ally in order to make usage of this ee t

7

,

8

,

9

.

3.2.8 Audibility of phase

Helmholtz and Ohm stated that the per eived olor of a tone of a omplex sound depends only on the

amplitude ratios but is independent of the phase spe trum. Indeed for most signals the phase shift of

the reprodu ing system doesn't inuen e the aural impression at all. However there are some ex eptions

that show that humans are not stri tly phase deaf. For instan e the masking ee t of low frequen y

tones depends not only on the amplitude spe trum but on the time fun tion as well

10

. An other phase

sensitive example signal is a omplex sound with many harmoni s with spe i phase relations:

s(t) =

∞∑

i=1

giicos(2π · i · f0 · t+ φi) (3.9)

The omparison between an in-phase version of s and one with random phase reveals learly audible

dieren es. For the in-phase version the following parameter setting is used: f0 = 100 Hz, gi = 1 and

φi = (i − 1)π/2. For the random phase version f0 and gi are identi al while for ea h i φi is set to a

random number between 0 and 2π 11

.

7

Erik Larsen, Ronald M. Aarts, Reprodu ing Low-Pit hed Signals through Small Loudspeakers, Journal of the Audio

Engineering So iety, vol. 50, no. 3, p.147-164 (2002).

8

Nay OO, Woon-Seng Gan, Mal olm Hawksford, Journal of the Audio Engineering So iety, vol. 59, no. 11, p.804-824

(2011).

9

Hao Mu, Woon-Seng Gan, Per eptual Quality Improvement and Assessment for Virtual Bass Systems, Journal of the

Audio Engineering So iety, vol. 63, no. 11, p.900-913 (2015).

10

E. Zwi ker, M. Zollner, Elektroakustik, Springerverlag, 1987, p. 250.

11

M. Laitinen, S. Dis h, V. Pulkki, Sensitivity of Human Hearing to Changes in Phase Spe trum, Journal of the Audio

Engineering Soei ty, vol. 61, p. 860-877 (2013)

77

3.2.9 Methods to al ulate and measure the loudness

For the determination of the loudness of stationary signals a standardized method - originally developed

by Zwi ker - exists

12

. The al ulation needs the third o tave band spe trum of the signal as input.

Re ently, loudness level meters have been developed that an even handle time varying sounds.

3.2.10 Nonlinear distortions of the ear

The transmission of the movement of the tympani membrane to the inner ear is not a perfe tly linear

pro ess. Thereby nonlinear distortions o ur. They manifest as sum- and dieren e tones if two tones

of dierent frequen ies are presented. The dieren e tones are espe ially disadvantageous as they are

not masked by the original tones. The strength of the most important dieren e tone at the frequen y

f = f2− f1 (where f2 is the frequen y of the higher stimulus tone and f1 orresponds to the frequen y

of the lower stimulus tone) an be estimated by

13

:

L(f2 − f1) = L(f1) + L(f2)− 130 dB (3.10)

where

L(f2 − f1): level of a tone at frequen y f2 − f1, that leads to the same sensation as the dieren e

tone produ ed by the nonlinearity.

L(f1): level of the lower frequen y stimulus tone

L(f2): level of the upper frequen y stimulus tone

The summation in Eq. 3.10 has to be understood arithmeti ally. The stimulation of the ear with two

tones of L(f1) = L(f2) = 90 dB produ es a level of the dieren e tone of 50 dB.

3.3 Properties of the ear for non stationary signals

3.3.1 Loudness dependen y on the signal length

The hearing pro ess shows a ertain delay. Very short events are not per eived with full loudness. The

maximal loudness is per eived just after a few tenths of a se ond. For signals shorter than 100 ms the

per eived loudness is proportional to the signal length or signal energy. Figure 3.3 shows the relation

between signal duration and loudness

14

.

40

45

50

55

60

65

1 10 100 1000

T [m s]

LN

[P

ho

n]

Figure 3.3: Relation of the per eived loudness level LN and the signal duration T for a 2 kHz tone

burst of 57 dB.

12

ISO Norm 532 A ousti s - Method for al ulating loudness level. 1975.

13

E. Zwi ker, Psy hoakustik, Springer, 1982.

14

E. Zwi ker, Psy hoakustik, Springer, 1982.

78

3.3.2 Temporal masking

Similarly as stationary signals an mask other frequen y omponents, strong transient signals an mask

weaker signals in the temporal vi inity of the masker. As shown in Figure 3.4, the hearing threshold

is shifted upwards just a few millise onds before and some hundredths of a se ond after the masker

appeared or disappeared.

-50 0 50 100 150 0 50 100 150 200

0

20

40

60

masking

masker

heari

ng

th

resh

old

[d

B]

tp [ms] ta [ms]

Figure 3.4: Temporal masking ee t for a masker of 200 ms duration. The hearing threshold is shifted

upwards already shortly before the masker is dete ted. After the dis ontinuation of the masker the

hearing threshold returns to its normal level after some tenths of a se ond.

3.4 Binaural hearing: lo alization of sound sour es

Within ertain limits the auditory system is apable to lo alize sound sour es a ording to their

dire tion and their distan es. For example in a noisy environment a listener an on entrate on a

spe i speaker and suppress the unwanted sound to a ertain extent ( o ktail party ee t).

The lo alization of sound sour es is usually des ribed with help of a spheri al oordinate system with

its origin at the head's lo ation. The lo alization in the verti al plane (elevation of the sour e) is

based on monaural attributes. The lo alization in the horizontal plane on the other hand (azimuth of

the sour e) uses inter-aural attributes whi h means dieren es between the signals at the two ears. To

improve the lo alization, humans perform permanently little rotational movements with their heads

and evaluate the resulting small variations. These movements help to dis riminate between sour es

that lie in front of and sour es that are behind the listener. This information is not available in the

presentation of re ordings over headphones.

The information that is available to the auditory system is omposed of the signals at the two ear

drums. The ex itation of the eardrums depends on frequen y and the sound in ident dire tion. As a

rst approximation the problem an be formulated as dira tion pattern on a sphere. The transmission

system free-eld → ear drum is usually des ribed by the head related transfer fun tion HRTF. This

transfer fun tion depends on the dire tion of in iden e and varies to some extent from person to person

15

,

16

,

17

. Knowledge of the head related transfer fun tion is essential for the equalization of headphones

or in the ontext of virtual reality appli ations (auralisation

18

).

15

H. Moeller, M. F. Soerensen, D. Hammershoei, C. B. Jensen, Head-Related Transfer Fun tions of Human Subje ts,

Journal of the Audio Engineering So iety, May, n.5, vol 43, p.300-321 (1995).

16

A. S hmitz, M. Vorländer, Messung von Aussenohrstossantworten mit Maximalfolgen-Hadamard-Transformation und

deren Anwendung bei Inversionsversu hen, A usti a, vol. 71, p.257-268 (1990).

17

Corey I. Cheng, Gregory H. Wakeeld, Introdu tion to Head-Related Transfer Fun tions (HRTFs): Representations

of HRTFs in Time, Frequen y, and Spa e, Journal of the Audio Engineering So iety, vol. 49, p.231-249 (2001).

18

L. Savioja, et. al. Creating Intera tive Virtual A ousti Environments, Journal of the Audio Engineering So iety, vol.

47, p.675-705 (1999).

79

3.4.1 Lo alization in the horizontal lane

The lo alization in the horizontal plane is based on two attributes. If sound is in ident from a lateral

dire tion as shown in Figure 3.5, the two ear signals dier in amplitude and time of arrival. Maximal

dire tional resolution is a hieved for frontal sound in iden e. In this ase azimuth hanges in the order

of 1

an be dis riminated. Figure 3.6 and Figure 3.7 show how time and level dieren es at the

two ears are mapped onto dire tional information. Completely lateral dire tion is per eived for time

dieren es of 630 µs and level dieren es of 10 dB.

At lower frequen ies (below about 800 Hz but above about 80 Hz), the auditory system uses mainly

time dieren es, for higher frequen ies (above about 1600 Hz), mainly level dieren es are evaluated

19

. For frequen ies in between, both attributes play a role.

φ

Figure 3.5: Dire tion φ of sound in iden e for the hearing in the horizontal plane.

−1.5 −1 −0.5 0 0.5 1 1.5

Dt [ms]

−90

−45

0

45

90

perceiv

ed

dir

ecti

on

[°]

Figure 3.6: Dete tion of the angle of in iden e φ in the horizontal plane in relation to the inter-aural

time dieren e ∆t.

3.4.2 Lo alization in the verti al plane (elevation)

For a sound sour e lo ated in front of the head but at dierent elevation angles, the two ear signals

don't dier at all. In this ase, no binaural attributes an be evaluated. The only information available

is the hange of the amplitude response of the HRTF in relation to the elevation angle. The elevational

resolution that an be a hieved depends strongly in the signal type and lies in the order of 10

. . .45.

3.4.3 Per eption of distan es

Up to a ertain degree the auditory system an estimate the distan e of a sound sour e. The most

important attribute that is evaluated is the strength of the signal. The louder a signal is, the shorter

is the per eived distan e to the sour e. In rooms the amount of reverberant sound in relation to the

dire t sound an be evaluated additionally.

19

A tually level dieren es are evaluated over the whole auditory frequen y range. However in typi al situations no

signi ant level dieren es o ur at low frequen ies due to dira tion around the head. In near eld appli ations with

small distan es to the sour e level dieren es at the two ears an o ur due to dierent distan e ratios.

80

−15 −10 −5 0 5 10 15

DL [dB]

−90

−45

0

45

90

perceiv

ed

dir

ecti

on

[°]

Figure 3.7: Dete tion of the angle of in iden e φ in the horizontal plane in relation to the inter-aural

level dieren e ∆L.

3.4.4 E hoes and the pre eden e ee t

In a situation with dire t sound and a shortly delayed opy of it, the auditory systems tends to merge the

two signals to one impression and to lo alize on the signal that arrives rst. This property is denoted as

pre eden e ee t

20

,

21

. There are two limitations asso iated with the pre eden e ee t (Figure 3.8).

Firstly, the lo alization on the rst arriving signal takes pla e only if the sound pressure level of the

delayed signal is not more than 10 dB higher than the dire t sound. Se ondly, the delay must be smaller

than 30 to 50 ms, depending on the level dieren es. If the delay is larger than 50 ms, the se ond

signal is per eived as a separate omponent, as an e ho. E hoes are unfavorable in the sense that they

disturb ommuni ation and thus lower spee h intelligibility.

0 10 20 30 40 50 60

delay [ms]

0

2

4

6

8

10

12

am

plifi

cati

on

[d

B]

Figure 3.8: The pre eden e ee t o urs for delay and ampli ation ombinations that lie below the

urve.

3.5 Hearing damage

3.5.1 Me hanisms

A hearing damage an have two auses. A rst ause is a possible me hani al damage of the inner

ear by an intense boom event. A se ond reason is a permanent long term overload of the auditory

system by exuberant sound. In this ase the metabolism of the inner ear an be overstrained with the

20

Helmut Haas, Über den Einuss eines Einfa he hos auf die Hörsamkeit von Spra he, A usti a, vol. 1, no. 2 (1951).

21

The pre eden e ee t, Ruth Y. Litovsky et al., Journal of the A ousti al So iety of Ameri a, vol. 106, p.1633-1654

(1999).

81

onsequen e that the hair ells are not supplied properly and are dying o over time. As the sensitivity

of the ear is biggest around 4 kHz, hearing losses develop often in this frequen y range rst. Later the

ae ted region enlarges and will over the important range for ommuni ation. This is the moment

where the damage will be ome obvious.

A serious disease of the ear is the tinnitus. Hereby the patient per eives tones and noises that do not

exist. In fatal ases tinnitus an seriously disturb the ability to on entrate and to relax. Tinnitus an

have dierent auses. However, in many ases a noise indu ed hearing loss stands at the beginning.

Up to now there is no real treatment available.

3.5.2 Assessment of the danger for a possible hearing damage

The evaluation of a possible danger for a hearing damage is based on a dose measure. The relevant

fa tors are sound pressure and time. The dose orresponds to the produ t of the two fa tors. An

in rease of one fa tor an be ompensated by a redu tion of the other.

The assessment of impulsive sound is based on the sound exposure level SEL or event level LE ,measured over a period of 1 hour. The SUVA denes as a limit an SEL = LE = 120 dB(A). The

single o urren e of a higher level may lead to a permanent damage of the ear. The ring of one shot

with an assault rie for example produ es an LE of 129 dB(A). In addition to the A-level LE riterion,

a maximum for the C-weighted peak level of 135 dB(C) has to be met.

For stationary noise SUVA has established the following limiting value: for permanent noise exposure

during 8 hours a day and 5 days a week the Leq must not ex eed 85 dB(A). In a year the assumed

working time sums up to 2000 hours. If the exposure o urs only during a portion of this time, higher

levels are tolerable (Table 3.2).

yearly time of exposure allowable Leq

2000 h 85 dB(A)

1000 h 88 dB(A)

500 h 91 dB(A)

250 h 94 dB(A)

Table 3.2: Allowable Leq values in dependen y of the yearly time of exposure a ording to the SUVA

limiting values.

A ording to today's knowledge, ultrasoni sound (20 kHz. . .100 kHz) doesn't ause harm if the un-

weighted maximum level is below 140 dB and the sound exposure level integrated over a period of 8

hours doesn't ex eed 110 dB. For infrasound (2 Hz. . .20 Hz) the orresponding limits are 150 dB for

the maximum level and 135 dB for the exposure level.

82

Chapter 4

Musi al Intervals

The o tave as a frequen y ratio 2:1 is the most fundamental musi al interval in western musi . The

equally tempered s ale in use today divides ea h o tave on a logarithmi basis in 12 half tones. Ea h

half tone orresponds thus to a frequen y ratio of 21/12 ≈ 1.059 ≈ 6%. The advantage of the equally

tempered s ale lies in the fa t that on a piano all intervals an be played starting from any half tone

and a ertain interval always orresponds to the same frequen y ratio. The disadvantage on the other

hand is that besides the o tave no other perfe t whole-numbered interval an be played. A fth for

example whi h represents a ratio of 3:2 in just s ale has to be played as 7 half tones in the equally

tempered s ale, orresponding to a ratio of 1.498. The pure fourth stands for a frequen y ratio of

4:3. This has to be approximated by 5 half tones resulting in a ratio of 1.3348. The deviation of the

frequen y intervals for the equally tempered s ale ompared to the just s ale are so small, that the

pleasure of musi is not disturbed. As an overview Table 4.1 shows the intervals and the frequen y

ratios for the equally tempered s ale.

interval tone number of half tones frequen y ratio just s ale

perfe t unison 0 1.0000

minor se ond des 1 1.0595

major se ond d 2 1.1225

minor third es 3 1.1892 6:5 = 1.2000

major third e 4 1.2599 5:4 = 1.2500

perfe t fourth f 5 1.3348 4:3 ≈ 1.3333

augmented fourth s 6 1.4142

diminished fth ges 6 1.4142

perfe t fth g 7 1.4983 3:2 = 1.5000

minor sixth as 8 1.5874

major sixth a 9 1.6818

minor seventh b 10 1.7818

major seventh h 11 1.8878

perfe t o tave ' 12 2.0000 2:1 = 2.0000

Table 4.1: Musi al intervals for the equally tempered s ale, starting with the tone .

Alexander John Ellis proposed in 1875 a mu h ner partition than just half tones, labeled as ent. Cent

stands for hundred and signies a logarithmi partitioning of a half tone interval into 100 steps. An

o tave has 12 half tones and orresponds therefore to 1200 ents. A ent stands for a frequen y ratio

of

1200√2 ≈ 1.00057779. In other words one ent orresponds to a frequen y hange of 0,057779 %. In

general a frequen y ratio f2 to f1 orresponds C ent where

C = 1200 log2

(

f2f1

)

= 1200ln(

f2f1

)

ln(2)[Cent] (4.1)

The other way round, C ent orrespond to a frequen y ratio f2/f1 of

f2f1

= 2C

1200(4.2)

83

Chapter 5

Outdoor sound propagation

The simplest ase of a sound propagation situation is given by a point sour e radiating in all dire tions

with equal strength in an unbounded homogeneous medium at rest. The sound pressure at an arbitrary

re eiver position an be determined by taking into a ount the geometri al spreading and the frequen y

dependent air absorption. However, in real situations usually further inuen e fa tors have to be

onsidered. Firstly the medium is never unbounded. In many ases the sour e and/or the re eiver are

in the vi inity of the ground. This ground surfa e leads to a ree tion of the sound waves and in the

intera tion with the dire t sound to interferen e ee ts. Besides the ree tion at the ground, additional

ree tions at other obje ts su h as walls or building fa ades may o ur. Se ondly, the medium is

usually not at rest and not homogeneous. This leads to a refra tion of sound waves and in onsequen e

to urved propagation. Thirdly the sound propagation between the sour e and the re eiver may be

interrupted by obsta les su h as trees or walls. In this ase, damping and dira tion ee ts have to be

taken into a ount.

5.1 Basi equation

The al ulation of an outdoor sound propagation problem is usually based on an equation in form of

Eq. (5.1). The relevant variables are the sour e strength - spe ied as a sound power level, a possible

orre tion for the dire tivity and a sum of attenuation terms

1

.

Lp(re eiver) = LW +D −∑

A (5.1)

where

Lp(re eiver): sound pressure level at the re eiver

LW : sound power level of the sour e

D: dire tivity of the sour e

A: attenuation during propagation

As most attenuations A are frequen y dependent, the al ulation a ording to Eq. 5.1 has to be

performed for dierent frequen y bands. Therefore the sound power is split into third-o tave or o tave

bands, then the propagation attenuation is al ulated for ea h band and nally the sound pressure

values at the re eiver for ea h band are summed up to a total level. For distin t lasses of noise sour es

with a dened spe trum, approximations for the A-weighted may be applied.

5.2 Dire tivity of the sour e

The simplest model of a sour e assumes equal radiation in all dire tions. Su h a hara teristi s is

denoted as omnidire tional or spheri al. If su h an omnidire tional sour e is lo ated lose to a ree ting

surfa e, the radiation is restri ted to a limited solid angle, leading to an ampli ation in these dire tions.

Table 5.1 lists the orresponding dire tivity values D from Eq. 5.1 for dierent ongurations of the

sour e.

In some ases the sour e itself an show a dire tivity with stronger radiation in some dire tions.

1

ISO Standard 9613: A ousti s - Attenuation of sound during propagation outdoors, Part 2: General method of

al ulation.

84

sour e onguration solid angle D[dB

open spa e 4π 0

in front of a surfa e 2π +3

in front of two orthogonal surfa es π +6

in front of three orthogonal surfa es ( orner)

π2 +9

Table 5.1: Dire tivity orre tions D in Eq. 5.1 for a point sour e in front of ree ting surfa es.

5.3 Attenuation terms

5.3.1 Geometri al divergen e

The geometri al divergen e is independent of frequen y and des ribes the redu tion of intensity or

sound pressure with distan e due to the distribution of the sound power on an area that in reases with

distan e. For an omnidire tional point sour e, the intensity on a spheri al surfa e around the sour e is

given by Eq.5.2.

I =W

4πd2(5.2)

where

I: intensity at distan e d from the sour e

W : sound power

For distan es larger than a few wavelengths, the ratio of sound pressure and sound parti le velo ity

equals the free eld impedan e and therefore

I =p2rms

ρ0c(5.3)

and

p2rms =Wρ0c

4πd2(5.4)

In the dB s ale the geometri al divergen e Adiv is given as (with the onversion onstant from sound

power level to sound pressure level in 1 m distan e)

Adiv = 20 log

(

d

d0

)

+ 11 [dB (5.5)

where

d: distan e sour e - re eiverd0: referen e distan e = 1 m

5.3.2 Atmospheri absorption

During sound propagation, a ertain fra tion of the sound energy is onverted into heat. Per unit

distan e the fra tion of absorbed energy is onstant. Translated into the dB s ale this orresponds to

(Eq. 5.6).

Aatm = αd [dB (5.6)

Atmospheri absorption is inuen ed by air temperature and humidity and depends strongly on

frequen y. For that reason the al ulation of air absorption should preferably be done in third o tave

bands. Table 5.2 shows the atmospheri absorption in dB/km for some temperature / humidity

ombinations. The values orrespond to the parameter α in Eq. 5.6 if the distan e d is inserted in km 2

.

2

ISO Norm 9613-1: A ousti s - Attenuation of sound during propagation outdoors.

85

temp [

C rel. humidity[% 63 125 250 500 1k 2k 4k 8k

10 70 0.1 0.4 1.0 1.9 3.7 9.7 32.8 117

20 70 0.1 0.3 1.1 2.8 5.0 9.0 22.9 76.6

30 70 0.1 0.3 1.0 3.1 7.4 12.7 23.1 59.3

15 20 0.3 0.6 1.2 2.7 8.2 28.2 88.8 202

15 50 0.1 0.5 1.2 2.2 4.2 10.8 36.2 129

15 80 0.1 0.3 1.1 2.4 4.1 8.3 23.7 82.8

Table 5.2: Coe ient α of atmospheri absorption in dB/km as a fun tion of pure tone frequen y for

dierent ombinations of temperature and humidity.

Table 5.2 shows a very strong in rease of the atmospheri absorption towards higher frequen ies.

Further away form a sour e, only the low frequen y omponents are audible.

The oe ients α of atmospheri absorption an be al ulated with the following set of formulas:

α = 8.686f2

([

1.84× 10−11

(

papr

)−1(T

T0

)1/2]

+

(

T

T0

)−5/2

×

×

0.01275

[

exp

(−2239.1

T

)][

frO +

(

f2

frO

)]−1

+

+ 0.1068

[

exp

(−3352.0

T

)][

frN +

(

f2

frN

)]−1)

(5.7)

frO =papr

(

24 + 4.04× 104h0.02 + h

0.391 + h

)

(5.8)

frN =papr

(

T

T0

)−1/2(

9 + 280h exp

−4.170

[

(

T

T0

)−1/3

− 1

])

(5.9)

h = hrpsat/prpa/pr

(5.10)

psat/pr = 10−6.8346(273.16/T )1.261+4.6151(5.11)

where

α: oe ient of air absorption in [dB/m

f : frequen y in [Hz

pa: air pressure in [kPa

pr: referen e air pressure = 101.325 kPa

T : air temperature in Kelvin

T0: referen e air temperature = 293.15 K

hr: relative humidity of the air as per entage (0 < hr < 100)

5.3.3 Ground ee t

In many pra ti al ases sound propagates lose to the ground. For larger distan es and small sour e and

re eiver heights the angles of in ident relative to the ground are almost grazing. In this situation the

dire t sound interferes with a signi ant ree tion from the ground. The frequen ies that are amplied

or attenuated depend on the sound path length dieren es and the phase shift at the ree tion. The

modi ation of the sound pressure at a re eiver due to the presen e of the ground is alled ground

ee t. An exa t analyti al solution an be given for simple geometries only (see below). However for

a broad band signal the A-weighted ground ee t an be estimated a ording to Eq. 5.12.

Aground = 4.8− 2hmd

(

17 +300

d

)

≥ 0 [dB(A) (5.12)

where

hm: average height above ground of the dire t sound propagation path [m

86

d: distan e sour e - re eiver [m

If the ground ee t al ulation is performed based on Eq. 5.12, an additional dire tivity orre tion DΩ

that in reases the re eiver level has to applied as:

DΩ = 10 log

(

1 +d2p + (hs − hr)

2

d2p + (hs + hr)2

)

(5.13)

where

hs: height of the sour e above ground [m

hr: height of the re eiver above ground [m

dp: sour e-re eiver distan e proje ted onto the ground plane [m

For at terrain the ISO standard 9613-2

3

des ribes a more subtle algorithm that yields the ground

ee t in o tave bands.

In ase of a point sour e above at homogeneous ground, the ground ee t an be al ulated exa tly

in the sense of an numeri al approximation to the wave theory

4

. Thereby lo ally rea ting ground is

assumed whi h means the boundary ondition at the ground is dened as the frequen y dependent

ratio of sound pressure and the normal omponent of the sound parti le velo ity (ground impedan e).

The al ulation is based on the following variables (see Fig. 5.1):

d: horizontal distan e sour e - re eiverhs: sour e height above groundhr: re eiver height above groundZ: impedan e of the ground, normalized to ρcR1: distan e sour e - re eiver

R2: distan e sour e - point of ree tion - re eiver

λ: wave lengthk: wave number =

2πλ

hs

h rS

R

d

R1

R2Z

ΨΨ B

Figure 5.1: Situation of a point sour e S above homogeneous ground with impedan e Z, B is the

ree tion point, R is the re eiver.

As already insinuated in Fig. 5.1, the sound pressure p(R) at the re eiver is omposed of two ontri-

butions: the dire t sound and the ground ree tion. In omplex writing p(R) an be stated as:

p(R) =1

R1ejkR1 +Q

1

R2ejkR2

(5.14)

where

Q: spheri al wave ree tion oe ient

The spheri al wave ree tion oe ient Q an be dedu ed from the plane wave ree tion oe ient

rp as

3

ISO Standard 9613: A ousti s - Attenuation of sound during propagation outdoors, Part 2: General method of

al ulation.

4

C. I. Chessel, Propagation of noise along a nite impedan e boundary. J. A ousti al So iety of Ameri a, 62, p.825-834

(1977).

87

Q = rp + (1− rp)F (w) (5.15)

where

rp =sin(ψ)− 1

Z

sin(ψ)+ 1Z

w = 1+j2

√kR2

(

sin(ψ) + 1Z

)

The fa tor F (w) in Eq. 5.15 an be approximated as

5

F (w) = 1 + j√πwe−w

2

erf (−jw) = 1 + j√πwwofz(w) (5.16)

The fun tion erf (−jw) in Eq. 5.16 denotes the omplex error fun tion

6

. For the evaluation of the

fun tion wofz(w) = e−w2

erf (−jw), a very e ient algorithm is available

7

,

8

.

The impedan e Z of the ground is frequen y dependent. Very often, the hara terization is based on

a one parameter model with the ow resistivity σ as variable. With help of the empiri al model by

Delany and Bazley

9

(5.17) the impedan e normalized to ρc an be al ulated for all frequen ies f .It should be noted that the sign of the imaginary part of the impedan e in Eq. 5.17 depends on the

onvention of the time dependen y in the omplex representation. A positive imaginary part as shown

here, assumes e−jωt 10

.

Z = 1 + 9.08

(

f

σ

)−0.75

+ j11.9

(

f

σ

)−0.73

(5.17)

where

Z: impedan e normalized to ρcf : frequen y [Hz

σ: ow resistivity [kPa·s/m2.

Table 5.3 shows orresponding ow resistivities for dierent ground types. Figure 5.2 demonstrates

exemplarily the frequen y response of the impedan e for lawn (σ = 300 kPa·s/m2).

Fig. 5.3 shows the frequen y responses of the ground ee t, al ulated with Eq. 5.14 for dierent

situations. For that purpose the resulting sound pressure at the re eiver is referen ed to the dire t

sound pressure. For grassy ground an ampli ation at very low frequen ies and an attenuation in the

mid frequen y range is very typi al.

ground type ow resistivity σ [kPa·s/m2

asphalt, water 20'000

hard natural ground 5'000

plow soil, gravel 500

lawn 300

grass land 150

hard snow 40

powder snow 10

Table 5.3: Flow resistivity for dierent ground types.

5

C. F. Chien, W. W. Soroka, A note on the al ulation of sound propagation along an impedan e surfa e. J. Sound

and Vib. vol. 69, 340-343 (1980).

6

M. Abramowitz, J. A. Stegun, Handbook of Mathemati al Fun tions.

7

W. Gauts hi, E ient Computation of the Complex Error Fun tion. SIAM J. Numer. Anal. vol. 7, 187-198 (1970).

8

Colle ted Algorithms from CACM. Algorithm 363.

9

M. E. Delany, E. N. Bazley, A ousti al properties of brous absorbent materials. Applied A ousti s vol. 3, 105-116

(1970).

10

G. A. Daigle et al. Some omments on the literature of propagation near boundaries of nite a ousti al impedan e,

Journal of the A ousti al So iety of Ameri a, vol. 66, p.918-919 (1979).

88

10 100 1000 10000

frequency [Hz]

0

20

40

60

80

100

120

140

160

norm

alized im

pedance

Re

Im

Figure 5.2: Real- and imaginary part of the impedan e of lawn, normalized with ρc

100 158 251 398 631 1000 1585 2512 3981 6310

third octave band frequency [Hz]

−40

−30

−20

−10

0

10

gro

und e

ffect

[dB

]

20 m

50 m

200 m

100 158 251 398 631 1000 1585 2512 3981 6310

third octave band frequency [Hz]

−40

−30

−20

−10

0

10

gro

und e

ffect

[dB

]

20 m

50 m

200 m

Figure 5.3: Ground ee t for lawn (left) and snow overed ground (right) with sour e and re eiver 1 m

above ground and distan es of 20, 50 and 200 m. The wide dip in the low frequen ies in ase of snow

explains the experien e that many transportation noise sour es are quieter during periods of snow.

5.3.4 Obsta les

Vegetation

Sound is attenuated while passing dense vegetation. This is mainly due to s attering ee ts at trunks

and bran hes. However, signi ant attenuation is found only for extensions of more than about 20

meters. One row of trees or bushes has no dire t ee t. Though a se ond order ee t is the fa t

that vegetation loosens the ground and by this redu es the ow resistivity whi h in turn inuen es the

ground ee t. An additional ee t of vegetation is the interruption of view whi h may be bene ial

from a psy hologi al point of view in noise abatement appli ations.

Table 5.4 shows the average attenuation Afoliage in o tave bands asso iated with dense vegetation. The

ee tive distan e is the sound path that passes through the vegetation.

ee tive distan e 63 125 250 500 1k 2k 4k

10. . .20m 0dB 0dB 1dB 1dB 1dB 1dB 2dB

20. . .200m 0.02dB/m 0.03dB/m 0.04dB/m 0.05dB/m 0.06dB/m 0.08dB/m 0.09dB/m

> 200m 4 dB 6 dB 8 dB 10 dB 12 dB 16 dB 18 dB

Table 5.4: Attenuation due to vegetation Afoliage as a fun tion of frequen y.

89

Noise barriers

Massive obsta les that interrupt the line of sight between sour e and re eiver lead to a signi ant

attenuation. In the ontext of road and railway noise, barriers are indeed a ommon approa h to

redu e the noise level at re eivers. Due to dira tion ee ts, still a relevant portion of the sound wave

an rea h the geometri al shadow zone behind an obsta le. This is due to the fa t that sound wave

lengths relevant for many noise sour es are in the same order of magnitude as typi al geometri al

dimensions. In the al ulation of the attenuation due to obsta les, the portion of sound energy that

goes through the obsta le is usually ignored. This is allowable in most ases if the area spe i mass

of the obsta le is larger than 10 kg/m

2.

The al ulation of the sound eld behind an innitely extended half plane is a lassi al task in theoreti al

a ousti s. Maekawa was the rst that dedu ed an empiri al formula for the barrier attenuation based

on theoreti al onsiderations and measurements in the laboratory. Hereby the attenuation is expressed

as a fun tion of one single parameter - the so alled Fresnel number N . N is dened as the ratio

N = 2z/λ where λ is the wave length and z is the dieren e of the path lengths around the obsta le

and through the obsta le.

Figure 5.4: Situation of a single obsta le between sour e and re eiver with the sound path around the

obsta le edge (d1, d2) and the path through the the obsta le (d).

The ISO standard 9613 al ulates the attenuation Ascreen due to a barrier as follows:

Ascreen = 10 log

(

3 +C2

λC3zKw

)

[dB (5.18)

where

C2 = 20

C3 = 1 for a single barrier

λ: wave length [m

z: dieren e of the path lengths around the obsta le and through the obsta le z = d1 + d2 − d (Fig.

5.4) [m

Kw: orre tion fa tor ≤ 1 to a ount for a redu ed attenuation ee t in ase of favorable propagation

onditions due to spe ial meteorologi al onditions (see below).

Remarks:

• If the obsta le just tou hes the line of sight between sour e and re eiver, the path length dieren e

z yields 0. The barrier attenuation a ording to Eq. 5.18 be omes 5 dB, independently of the

frequen y. If the obsta le height is lowered further, still a path length dieren e an be evaluated.

If the orresponding value is used with negative sign, the formula yields a smooth transition to

the ase where the barrier attenuation vanishes.

• As expe ted, Eq. 5.18 yields a barrier attenuation in the shadow zone that in reases with fre-

quen y.

• If a barrier attenuation is present, the attenuation by the ground Aground (ground ee t) should

be ignored.

A more a urate solution of the sound eld behind a barrier is given by Pier e

11

. The insertion loss

IL, that means the dieren e between the re eiver level with obsta le and the level without obsta le

an be al ulated as:

11

Allan D. Pier e, A ousti s, published by the A ousti al So iety of Ameri a (1991).

90

IL = −10 log

(

∣

∣

∣

∣

H(X)− ejπ4

√2AD(X)ej

π2X2

∣

∣

∣

∣

2)

[dB (5.19)

where

H(X): Heaviside fun tion, = 0 if the re eiver is in the geometri al shadow, = 1 in all other ases.

X =√

2kπ (L −R)

k: wave number = 2π fcf : frequen yc: speed of sound

L: path length from the sour e to the re eiver around the obsta le

R: path length from the sour e to the re eiver through the obsta le

AD(X): dira tion integral = f(X)− jg(X)f(X), g(X): auxiliary Fresnel fun tions, for whi h the following approximation exist:

f(X) ≈ 1+0.926X2+1.792X+3.104X2

g(X) ≈ 12+4.142X+3.492X2+6.67X3

In typi al outdoor noise ontrol appli ations - e.g. in the ontext of road tra noise - barrier

attenuations in the order of 5 to 15 dB an be a hieved. A barrier is most ee tive, if it is positioned

lose to the sour e or lose to the re eiver. As a onsequen e of turbulen e and inhomogeneities of the

air the maximum barrier attenuation is limited to 20. . .25 dB. During the installation of noise barriers

it has to be assured that no gaps o ur as they would lower the attenuation ee t onsiderably.

In some ases it is important that noise barriers are equipped with an absorbing surfa e to avoid

ree tions in the opposite dire tion. Methods to determine the hara teristi s of noise barriers in situ

are des ribed in the ISO standard ISO 10847: In-situ determination of insertion loss of outdoor noise

barriers of all types. An ex ellent overview of possible modi ations of the top se tion of noise barriers

to improve the attenuation ee t an be found in the paper by Ulri h

12

.

5.4 Ree tions

Besides the ground, additional surfa es and obje ts an ree t sound. They introdu e additional sound

propagation paths and thus rise the sound pressure at the re eiver. As the path lengths usually dier

signi antly from the dire t sound, the dierent ontributions an be summed up energeti ally. If the

ree ting obje t is a at surfa e, the ree tion an be dealt with the on ept of mirror sour es. The

riteria for the o urren e of spe ular ree tions are

• the point of ree tion lies on the ree ting surfa e

• the ree ting surfa e is large enough in relation to the sound wave length.

The test of a su ient ree tor size at the frequen y fc an be performed by he king if Eq. 5.20 is

fullled.

fc >2c

(lmin cos(β))2ds,odo,rds,o + do,r

(5.20)

where

c: speed of sound

ds,o: distan e sour e - point of ree tiondo,r: distan e point of ree tion - re eiver

β: angle of in iden e relative to the surfa e normal dire tion

lmin: smallest dimension of the ree tor

If the ree ting surfa e has absorbing properties, a orresponding attenuation has to be a ounted for.

If the ree ting obje t is not su iently at, the mirror sour e on ept an no longer be applied. The

handling of diuse ree tions is usually more di ult. As an example, Fig. 5.5 shows the ree tion at

12

S. Ullri h, Vors hläge und Versu he zur Steigerung der Minderungswirkung einfa her Lärms hutzwände, Strasse +

Autobahn 7, p.347-354 (1998).

91

a forest rim. Ea h tree s atters a ertain amount of sound energy. There is no sharp ree tion as in

ase of at surfa es but a sort of reverberation with a distin t temporal smearing.

0 0.1 0.2 0.3 0.4 0.5

time [s]

Figure 5.5: Level time urve of a gun shot ree ted at a forest rim. The dire t sound is followed by a

ree tion that is strongly smeared over time.

5.5 Meteorologi al ee ts

Up to now the medium air was assumed to be homogeneous, in rest and time invariant. All three

onditions are usually not fullled. Of importan e regarding possible sound propagation attenuation

variations are verti al temperature and wind speed gradients and the temporal and lo al inhomogeneities

in the air layer lose to the ground. Temperature and wind speed gradients lead to a urvature of the

propagation paths. Lo al inhomogeneities of the air produ e s attering ee ts.

5.5.1 Temperature gradients

The mass of the atmosphere generates an average pressure of 1013 hPa on sea level. With in reasing

height above ground, the pressure drops by about 12 Pa per meter. As a onsequen e of this pressure

de rease a pa ket of air that moves upwards ools down with about 1

per one hundred meters. A

temperature strati ation with a gradient of -1

/100m is alled adiabati strati ation.

The adiabati strati ation orresponds to the basi state of the atmosphere without additional

exterior inuen es. However, during day time with strong in oming sound radiation the ground and

with a ertain delay the air layer above is heated up. This leads to a strong negative temperature

gradient orresponding to de reasing temperature with in reasing height. This is alled an unstable

strati ation. On the other hand during nights with lear sky, the ground looses energy due to outgoing

radiation. This leads to a strong ooling of the ground and the adjoining layers of air. In the following,

a positive temperature gradient develops in the lowest few meters. This ondition is alled stable

strati ation or temperature inversion. It should be noted that a stable strati ation an only develop

if there are no strong winds.

In both regimes with unstable and stable strati ation the temperature gradients are largest lose to

the ground and be ome smaller with in reasing height. The temperature as a fun tion of height above

ground an be des ribed with an approa h as shown in Eq. 5.21.

T (z) = T (0) + kz0.2 (5.21)

where

T (z): temperature [

C at height z [m above the ground

k: onstant depending of the stability ondition with values k = −1.9 in the very unstable ase and

k = 2.6 for very stable onditions

92

Consequen es of temperature strati ation for the sound propagation

As the speed of sound depends on temperature, a temperature gradient leads to a gradient of the

ee tive propagation speed. A dire t onsequen e of this is a urvature of the sound paths. In ase of

unstable strati ation during day time the urvature points away from the ground (Figure 5.6). In larger

distan es shadow zones evolve with a orresponding strong attenuation. On the other hand during lear

nights with stable strati ation the sound speed in reases with height, leading to a urvature towards

the ground (Figure 5.7). This results in a lowering of the attenuation ompared to day time. It is even

possible that obsta les loose their ee t as they are surmounted by the propagation path.

shadow

zone

shadow

zone

Figure 5.6: Curvature of sound rays due to a negative temperature gradient. In larger distan es a

shadow zone develops where the sound pressure is strongly attenuated.

Figure 5.7: Curvature of sound rays due to a positive temperature gradient.

5.5.2 Wind

Due to fri tion in the vi inity of the ground, wind speed shows always a verti al gradient. The wind

speed prole u(z) an be des ribed with help of Eq. 5.22

13

.

u(z)

uref=

(

z − d0zref − d0

)α

(5.22)

where

u(z): average wind speed [m/s at the height z [m above ground

uref : average wind speed at the referen e height zref [m above ground (typ. 10 m)

d0: oset height [m, situation dependent a ording to Table 5.5

α: prole exponent, situation dependent a ording to Table 5.5

site d0 [m αwater, i e or snow 0 0.08. . . 0.12

gras land 0 0.12. . . 0.18

parks, agglomeration 0.75h 0.18. . . 0.24

forests, urban areas 0.75h 0.24. . . 0.40

Table 5.5: Oset heights and prole exponents for Eq. (5.22). The parameter h orresponds to the

average height of buildings and / or vegetation [m.

Consequen es of wind regarding sound propagation

The sound propagation in a moving medium has to onsider the sound speed ve tor (normal to the

wave front) and the velo ity ve tor of the medium. The wave front at time t+∆t an be found from

the front at time t by ve tor addition of the sound speed ve tor and the medium velo ity ve tor (Fig.

5.8).

13

VDI-Ri htlinie 3782, Blatt 12: Umweltmeteorologie, Physikalis he Modellierung von Strömungs- und Ausbre-

itungsvorgängen in der atmosphäris hen Grenzs hi ht (1999).

93

c

v

ray

Figure 5.8: The momentary propagation speed of a point on a wave front is given by addition of the

sound speed ve tor ~c (normal to the wave front) and the medium velo ity ve tor ~v.

The important inuen e of wind on sound propagation is a result of the verti al wind speed gradient.

In downwind dire tion sound propagates faster with in reasing height. Similarly as in ase of stable

temperature strati ations, sound propagates no longer along straight lines but be omes a urvature

towards the ground. In the upwind dire tion the urvature points upwards (Figure 5.9.

Figure 5.9: Curvature of sound rays due to a wind speed gradient. In the upwind dire tion a shadow

zone develops where the sound pressure is strongly attenuated.

5.5.3 Favorable and unfavorable sound propagation onditions

The inuen e of wind and temperature gradients on sound propagation an be divided roughly into the

two ategories favorable and unfavorable sound propagation onditions. Favorable onditions are given

if the propagation urvature is oriented towards the ground, unfavorable onditions are en ountered in

ase of a bending upwards.

For engineering appli ations, the propagation onditions are usually spe ied in four lasses

14

:

• M1: unfavorable sound propagation onditions

• M2: neutral onditions (no bending)

• M3: favorable sound propagation onditions

• M4: very favorable sound propagation onditions

The propagation lasses are determined by the temperature strati ation and the omponent of the

wind speed v in propagation dire tion. A spe i meteorologi al situation an be mapped onto the

orresponding propagation lass as shown in Table 5.6.

v < -1 -1 < v < 1 1 < v < 3 3 < v < 6 v > 6

over ast sky M1 M2 M3 M3 M4

lear sky during day M1 M1 M2 M3 M4

lear sky during night M1 M4 M4 M4 M4

Table 5.6: Mapping of a meteorologi al situation onto the propagation lasses M1 to M4. v is the windspeed omponent proje ted onto the propagation dire tion from sour e to re eiver in m/s.

14

ISO 1996-2rev, A ousti s - Des ription, measurement and assessment of environmental noise - Part 2: Determination

of environmental noise levels

94

For the distin tion between over ast/ lear sky, the following riteria an be used:

primary riteria

• lear during night: if temperature dieren e measured at 2.0 m and 0.05 m above ground is larger

than 1.5

C

• lear during day: if global radiation > 200 W/m

2

alternative riteria

• lear during day/night: if daily (24h) temperature variation at 2.0 m above ground is larger than

10

C

• lear during day/night: if loud overage < 4/8

5.5.4 Turbulen e

Wind ow over non-at terrain or lo ally varying heating of the ground surfa e lead to inhomogeneities

of the air in the surfa e layer. These inhomogeneities are alled turbulen e. Turbulen e is responsible

for arbitrary variations of the propagation attenuation between sour e and re eiver. However more

important are s attering ee ts that an ree t sound energy into geometri al shadow zones and the

ee t of de orrelation between dire t and ground ree ted sound. The in orporation of turbulen e into

al ulation s hemes an be done in dierent ways as e.g. des ribed here

15

.

5.5.5 Cal ulation of meteorologi al ee ts on sound propagation

The inuen e of meteorologi al ee ts on sound propagation an be onsidered in dierent ways.

Empiri al orre tions of barrier attenuation The possible variation of the propagation attenuation

due to meteorologi al ee ts is espe ially large in ase of an obsta le between sour e and re eiver.

For downwind onditions or for stable strati ation the barrier attenuation an be signi antly

redu ed. There are barrier attenuation formulas su h as ISO 9613-2 with empiri al orre tions

for favorable propagation onditions.

Analyti al solutions of sound ray paths Under the assumption of linear verti al proles of the ee -

tive sound speed ( onstant gradient), the urvature of the sound rays an be des ribed analyti ally.

The resulting rays are ir les. They an be onstru ted for arbitrary sour e and re eiver positions

and the onsequen es for a barrier attenuation or the ground ee t an be al ulated easily

16

.

Ray tra ing With ray tra ing al ulation s hemes

17

, the propagation of sound rays an be determined

for arbitrary ee tive sound speed proles (Fig. 5.10). Sound pressure levels at a re eiver point

an be determined by evaluating the density of the rays.

Numeri al solutions of the wave equation Several strategies are known to nd approximate numer-

i al solutions of the wave equation. As the distan es between sour e and re eiver are usually large,

lassi al methods su h as Finite Elements are out of question due to the exploding al ulation

eort. More suitable are approximations su h as the Paraboli Equation (PE) that assume pure

forward propagating waves and yield a numeri al solution of the wave equation. The benet of

the onstraint of forward propagation is that fa t that a stepwise solution of small systems of

equations is possible

18

.

15

P. Chevret et al. A numeri al model for sound propagation through a turbulent atmosphere near the ground. J.

A ousti al So iety of Ameri a, vol. 100, p.3587-3599 (1996).

16

A. L'Esperan e et al., Heuristi Model for Outdoor Sound Propagation Based on an Extension of the Geometri al

Ray Theory in the Case of a Linear Sound Speed Prole, Applied A ousti s, vol. 37, p. 111-139 (1992).

17

Robert J. Thompson, Ray-a ousti intensity in a moving medium, Journal of the A ousti al So iety of Ameri a, vol.

55, p. 729-737 (1974).

18

Erik M. Salomons, omputational atmospheri a ousti s, Kluwer A ademi Publishers, 2001.

95

Figure 5.10: Example of a ray tra ing simulation for downwind of 5 m/s at a height of 10 m above

ground. The horizontal axis is the oordinate in propagation dire tion, the verti al axis is the height

above ground (note the dierent s aling of the axis). The rays start at the sour e on the left. They are

bent downwards and an thus surmount obsta les. At ertain points rays interse t. In these so alled

austi s the energy density be omes innitely high whi h an obviously not be true. Within the ray

tra ing model, no statement about the sound pressure in these points is possible.

96

Chapter 6

Absorption and ree tion

If a sound wave hits a boundary surfa e, only a portion of the in oming energy is ree ted

1

. The

energy that is not ree ted splits into a portion that is absorbed and a portion that is transmitted. The

absorbed energy is onverted into heat. The transmission is the result of ex itation of the boundary to

vibrations and then as a onsequen e sound is radiated on the rear side. Often the transmitted portion

is not addressed expli itly whi h means that this ontribution is added to the absorbed portion.

6.1 Chara terization

The quantitative des ription of the property of a surfa e to absorb or ree t sound uses the absorption

oe ient or the ree tion oe ient. The absorption oe ient α is dened as the ratio of the

energies of absorbed and in ident sound:

α =absorbed energy

in ident energy

(6.1)

The ree tion oe ient R on the other hand is the ratio of the sound pressures of ree ting and

in oming sound:

R =sound pressure of ree ted wave

sound pressure of in ident sound wave

(6.2)

The absorption oe ient is a real number in the range 0 . . . 1. The ree tion oe ient is a omplex

number and des ribes the amplitude ratio and the phase shift during ree tion. Under the assumption

that the whole in ident energy splits into absorption and ree tion, a relation between α and R an be

established:

α = 1− |R|2 (6.3)

6.2 Types of absorbers

6.2.1 Porous absorbers

Porous absorbers are usually made from glass bers or organi bers or open foam. They fun tion as

absorbers due to fri tion losses when the air moves ba k and forth in the pores. The relevant sound eld

variable is thus the sound parti le velo ity. Consequently the optimal positioning of porous absorbers is

at lo ations with high sound parti le velo ity. It is therefore bene ial to install a porous absorber with

a ertain distan e to an a ousti ally hard boundary.

6.2.2 Resonan e absorbers of type Helmholtz

Helmholtz resonan e absorbers are formed by an a ousti al spring and an a ousti al mass. The spring

is realized by a ompressible volume of air, while the mass orresponds to a olumn of air that an be

a elerated (Fig. 6.1).

1

T. J. Cox, P. D'Antonio, A ousti Absorbers and Diusers, Taylor and Fran is, 2009.

97

S

V

l

Figure 6.1: Resonan e absorber of type Helmholtz. The mass and spring are realized by a olumn of

air ( ross se tion S and length l) and a volume V .

The resonan e frequen y of a mass/spring system with mass m and stiness s is

f0 =

√

sm

2π(6.4)

The mass m is given by the mass in the ylinder and a portion of vibrating air at the end of the ylinder.

This additional mass is introdu ed in the al ulation as a mouth orre tion. With ρ as density of air,

the moving mass is:

m = ρ(l + lcorr)S (6.5)

The mouth orre tion an be approximated as lcorr ≈ 0.8R where R orresponds to the radius of the

ylinder

2

.

The stiness s of the spring an be determined with help of the Poisson law (Eq. 1.14) for adiabati

pro esses:

s = c2ρS2

V(6.6)

where c is the speed of sound. Finally the resonan e frequen y f0 is found as

f0 =c

2π

√

S

V (l + lcorr)(6.7)

Without further measures the frequen y urve of absorption shows a large peak in a narrow band

only. The absorption ee t an be enlarged over a wider frequen y range by introdu ing damping (an

a ousti al resistan e su h as porous material) at the position of the ne k where the sound parti le

velo ity is highest.

There are dierent realizations of Helmholtz resonators. A rst possibility is a stru ture that onsists

of a layer of damping material and a plate with holes or slits on top of it. The air in the holes or slits

a ts as an a ousti al mass, the air in the damping material is the a ousti al spring.

An other version uses a sheet of metal that is installed in a ertain distan e to the wall or eiling. In

this ase the a ousti al mass is dominated by the ne k orre tion. The damping is usually realized

with a thin tissue mounted on the rear side of the metal sheet.

The extra damping material an be omitted if the holes have a very small diameter (< 1 mm)

3

,

4

,

5

.

Su h Helmholtz resonators are alled mi roperforated absorbers. The fri tion loss in the small holes is

large enough to realize su ient damping. It is thus possible to onstru t absorbers from one material

only. If this material is glass or a ryli glass, transparent absorbers are possible whi h opens very

2

The mouth orre tion l yields a non vanishing mass even if the length of the ylinder tends to 0.

3

Dah-You Maa, Mi roperforated-Panel Wideband Absorbers, Noise Control Engineering Journal, no. 3, vol. 29 (1987).

4

Einsatz mikroperforierter Platten als S hallabsorber mit inhärenter Dämpfung, H. V. Fu hs, X. Zha, A usti a, no. 2,

vol. 81 (1995).

5

I. Falsa, A. Ohadi, Design guide of single layer mi ro perforated panel absorber with uniform air gap, Applied

A ousti s, vol. 126, p.48-57 (2017).

98

interesting design possibilities. Alternative onstru tions use slits instead of holes

6

or thin layers of air

between two adja ent plates

7

or multi-layer arrangements

8

.

Figure 6.2 shows the performan e of absorption for two geometries of a perforated absorber. The smaller

the holes, the higher the damping and thus the broader the frequen y range with high absorption.

79 125 198 315 500 794 1260 2000 3175 5040 8000

frequency [Hz]

0

0.2

0.4

0.6

0.8

1

type 1

type 2

Figure 6.2: Cal ulated absorption as a fun tion of frequen y for normal sound in iden e. Absorber type

1: plate thi kness = 3 mm, hole diameter = 0.4 mm, spa ing between holes = 2 mm, distan e to wall

= 100 mm; absorber type 2: plate thi kness = 3 mm, hole diameter = 2 mm, spa ing between holes

= 15 mm, distan e to wall = 50 mm.

6.2.3 Membrane absorbers

Membrane absorbers or panel absorbers are an other realization of a spring - mass resonan e absorber.

In ontrast to Helmholtz absorbers the mass is realized by a thin plate or foil

9

. The spring is

determined by the stiness of the layer of air between the plate and the rigid wall. If foils are used as

mass, their stiness has to be added to the stiness of the air. Diaphragmati absorbers have to be

onstru ted as boxes to avoid that air an es ape at the edges.

As a resonan e ee t is responsible for the absorption, membrane absorbers are frequen y sele tive.

They are mainly used for low frequen y absorption. The resonan e frequen y f0 for whi h highest

absorption is obtained is given as

f0 =

√

s′′

m′′

2π(6.8)

where

s′′: stiness per unit aream′′

: mass per unit area

Similarly to the ase of Helmholtz absorbers the stiness is found as

s′′ =ρc2

lw(6.9)

where

lw: distan e of the panel to the rigid wall

6

R. T. Randeberg, Adjustable Slitted Panel Absorber, A ta A usti a united with A usti a, vol. 88, p.507-512 (2002).

7

R. T. Randeberg, A Helmholtz Resonator with a Lateral Elongated Ori e, a ta a usti a, vol. 86, p.77-82 (2000).

8

Y.J. Qian et al., Pilot study on wideband sound absorber obtained by adopting a serial-parallel oupling manner,

Applied A ousti s, vol. 124, p.48-51 (2017).

9

K. Sakagami et al., Sound Absorption of a Cavity-Ba ked Membrane: A Step Towards Design Method for Membrane-

Type Absorbers, Applied A ousti s, vol. 49, no. 3, pp. 237-247 (1996).

99

and nally

f0 =c√

ρm′′lw

2π(6.10)

By lling the volume of air between the panel and the rigid ba k wall with a porous material, the

absorption an be in reased and extended to a broader frequen y range.

For pra ti al appli ations, ertain onditions should be fullled. In general, best results are obtained

for large values of the distan e lw. However lw needs to be small ompared to the wave length λ0at the resonan e frequen y. Usually one tries to fulll the ondition lw < λ0/12. Further the panel

shouldn't be too small, a minimum area of 0.4 m

2is stipulated. In addition the proportions of the

panel shouldn't be too extreme, the minimum length of ea h panel side is 0.5 m.

Panel absorbers an be ombined with porous absorbers that are put on top. At low frequen ies where

the panel absorber is a tive, the porous absorber is transparent. However the additional mass of the

porous layer has to be onsidered.

6.3 Measurement of absorption and ree tion

6.3.1 Kundt's tube

The measurement in Kundt's tube allows for the determination of the absorption oe ient under

normal in iden e for relative small material probes

10

.

Kundt's tube serves to reate a one-dimensional plane wave sound eld at dis rete frequen ies (Figure

6.3). For that purpose a loudspeaker lo ated at one end of the tube generates a sine wave. This wave

propagates in the tube to the other end and will be ree ted at the hard termination. Thereby the

in ident and ree ted sound wave form an interferen e pattern with pressure maxima and minima.

By introdu ing absorbing material in front of the hard termination, the ree tion is redu ed and as a

onsequen e the sound pressure maxima de rease and the minima in rease. As will be shown below,

the absorption oe ient an be determined from the ratio of sound pressure in the maxima and

minima alone.

To guarantee that only plane waves along the tube axis o ur, the frequen y has to be limited to a

value su h that the orresponding wave length is larger than the diameter of the tube.

Figure 6.3: Kundt's tube with the loudspeaker at one end and the material probe in front of the hard

termination at the other end. In the enter of the tube diameter a probe mi rophone an be moved

along the tube axis to dete t sound pressure maxima and minima.

With pr as sound pressure of the wave ree ted at the end of the tube and pe as sound pressure of the

in ident wave one an write:

prpe

=√1− α (6.11)

The sound pressure maxima are formed by onstru tive interferen e of in ident and ree ted wave:

pmax = pe + pr = pe(1 +√1− α) (6.12)

10

ISO-Norm 10534: A ousti s - Determination of sound absorption oe ient and impedan e in impedan e tubes, Part

1: Method using standing wave ratio.

100

The sound pressure minima on the other hand result as destru tive interferen e between in ident and

ree ted wave:

pmin = pe − pr = pe(1−√1− α) (6.13)

With the ratio

n =pmax

pmin

(6.14)

the absorption oe ient an be al ulated as

α = 1−(

n− 1

n+ 1

)2

(6.15)

6.3.2 Impedan e tube

The measurement in the tube of Kundt is time onsuming, as for ea h frequen y the maxima and

minima have to be sear hed and evaluated. In this respe t the impedan e tube is a more elegant

method

11

. The geometry of loudspeaker, tube and probe is similar to Kundt's tube. However the

sound pressure is not observed along the tube axis but at two x positions. The ex itation is wide band

noise, allowing to extra t spe tral information with one single measurement. For a given geometry

(distan ed between the two mi rophones and distan es to the probe) the ratio between in oming and

ree ted wave an be evaluated by measuring the omplex transfer fun tion between the mi rophones

12

. From the omplex pressure ree tion fa tor the impedan e and the absorption oe ient an be

al ulated.

6.3.3 Reverberation hamber

The measurement of sound absorption in the reverberation hamber

13

is based on the inuen e of

absorption on the reverberation pro ess. After swit hing o a sound sour e in a room with hard

surfa es, the sound pressure doesn't drop to zero immediately. The sound waves are still ree ted

ba k and forth between the walls, oor and eiling. As they loose energy only slowly, the observable

reverberation pro ess an last for several se onds. The reverberation is des ribed by the reverberation

time T . The parameter measures the time for a de rease of the sound energy density to 1/1'000'000

of its initial value.

If sound absorbing material is introdu ed, the reverberation time de reases. The relation between

reverberation time T , room volume V and Absorption A an be expressed by the formula of Sabine:

T =0.16V

A(6.16)

From two measurements of T in the empty room and in the the room with absorbing material, the

in rease of absorption ∆A by the material an be determined. With knowledge of the area S of the

introdu ed material the absorption oe ient is found as αs = ∆A/S.

For maximum a ura y it is bene ial to aim at large dieren es between the empty room measure-

ment and the measurement with the material installed. For that reason reverberation hambers are

onstru ted with as less initial absorption as possible. The walls, the oor and the eiling are thus

made from a ousti ally hard materials. To redu e the tenden y of low frequen y resonan es, the walls

are usually oriented in su h a way that opposite walls are not in parallel. In addition, ree tors and

diusers may be installed in the room to improve the diusivity of the sound eld. The area of the

material probe lies usually between 10 and 12m

2.

The absorption oe ients αs determined in the reverberation hamber do not mat h exa tly with the

values found in Kundt's tube of the impedan e tube. One reason is the dieren e in the ex iting sound

11

ISO-Norm 10534: A ousti s - Determination of sound absorption oe ient and impedan e in impedan e tubes, Part

2: Transfer-fun tion method.

12

J. Y. Chung, D. A. Blaser, Transfer fun tion method of measuring in-du t a ousti properties. Journal of the

A ousti al So iety of Ameri a, vol. 68, p. 907-921, 1980.

13

ISO Norm 354 A ousti s - Measurement of sound absorption in a reverberation room. 1985.

101

eld. In the tubes only perpendi ular in iden e is investigated while in the reverberation hamber the

angles vary between 0

and 90

. In some ases αs values > 1 o ur, whi h doesn't make sense from

a physi al point of view. The reason for this is that important assumptions for the Sabine formula are

violated.

6.3.4 In situ measurement of impulse responses

In some ases it is not possible to put the stru ture or material of interest in the impedan e tube or

bring it to the reverberation hamber. Here in situ impulse response measurements in an appropriate

geometri al onguration may yield useful information. The loudspeaker - mi rophone - absorber

geometry has to be hosen in su h way that the dire t sound, the ree tion from the absorber and

other unwanted ree tions an be separated on the time axis. Two main di ulties are linked to the

problem of evaluating an absorption oe ient. To a ount for the dire t and ree ted sound path

length ratio, a normalization step is ne essary. This is easily done for at absorbers but an ause

major di ulties if the surfa e of interest is signi antly stru tured in depth. The se ond problem

arises from the requirements at low frequen ies. The evaluation of the low frequen y range makes

large dimensions of the absorber ne essary (see Fresnel zones).

For a re ent review of in situ absorption measurement te hniques see

14

.

6.4 Cal ulation of absorption and ree tion from impedan e

relations

6.4.1 Normal in iden e

A plane wave is onsidered that propagates in a medium with impedan e Z0. The medium is bounded

by a medium with impedan e Z1. It is assumed that the wave hits the impedan e dis ontinuity

Z0 → Z1 perpendi ularly.

The in ident sound wave has sound pressure pI and sound parti le velo ity vI with

pIvI

= Z0 (6.17)

The ree ted wave has sound pressure pII and sound parti le velo ity vII where

pIIvII

= Z0 (6.18)

At the surfa e of the medium Z1, sound pressure and sound parti le velo ity add up to

15

p = pI + pII

v = vI − vII (6.19)

with the ondition:

p

v= Z1 (6.20)

From

pI + pII = Z1

(

pIZ0

− pIIZ0

)

(6.21)

follows nally

pIIpI

= R =Z1 − Z0

Z1 + Z0(6.22)

14

E. Brandao, A. Lenzi, S. Paul, A Review of the In Situ Impedan e and Sound Absorption Measurement Te hniques,

A ta A usti a united with A usti a, vol. 101, p. 443-463, 2015.

15

The s alar quantity sound pressure adds up with a positive sign, while the ve tors sound parti le velo ity adds up

with negative sign due to reversed orientation.

102

Eq. 6.22 demonstrates that the ree tion fa tor R approa hes 1 for in reasing dieren e of Z1 and

Z0. On the other hand, maximum absorption will show for Z1 = Z0. An absorber is hara terized by

the property that it doesn't introdu e a signi ant resistan e to the in oming wave.

If a layer of porous absorption is pla ed in front of a hard wall, the resulting impedan e is in reased

ompared to the impedan e of the absorber itself. As a rule of thumb the thi kness of the absorber

should be larger than a quarter of the wave length of the lowest frequen y that should be absorbed.

6.4.2 Oblique in iden e

For many materials it an be assumed (as a rst order approximation) that the propagation in the

material itself is perpendi ular to the surfa e due to refra tion at the entry of the oblique in ident wave.

In this ase the rea tion of the material at any point is independent of the rea tion at any other point,

whi h is alled lo al rea tion. With this assumption one nds

pIIpI

= R =Z1 − Z0

cos(φ)

Z1 +Z0

cos(φ)

(6.23)

with

φ: angle of in ident and outgoing wave relative to the surfa e normal dire tion

The nominator in Eq. 6.23 an be ome 0 also for Z1 > Z0 by adjustment of φ. This means that for any

impedan e dis ontinuity Z0 → Z1 perfe t absorption is a hieved for a ertain angle of in iden e. In the

extreme ase of φ→ 90 the ree tion fa tor R approa hes -1, independently of Z1. This orresponds

to total ree tion with a phase shift of 180

.

6.5 Typi al values of absorption oe ients

There exist olle tions of data of absorption oe ients for dierent materials

16

. Usually o tave band

values of αs measurements in the reverberation hamber are shown. The following gures give a little

overview.

parquet oor wooden stage

0

0.2

0.4

0.6

0.8

1

125 250 500 1000 2000 4000

frequency [Hz]

ab

sorp

tio

n c

oeff

icie

nt

0

0.2

0.4

0.6

0.8

1

125 250 500 1000 2000 4000

frequency [Hz]

ab

sorp

tio

n c

oeff

icie

nt

16

Fasold, Sonntag, Winkler, Bau- und Raumakustik, Verlagsgesells haft Rudolf Müller, Köln-Braunsfeld, 1987.

103

stone oor arpet, thi kness 5 mm

0

0.2

0.4

0.6

0.8

1

125 250 500 1000 2000 4000

frequency [Hz]

ab

sorp

tio

n c

oeff

icie

nt

0

0.2

0.4

0.6

0.8

1

125 250 500 1000 2000 4000

frequency [Hz]

ab

sorp

tio

n c

oeff

icie

nt

plaster a ousti ally optimized plaster, thi kness 20

mm

0

0.2

0.4

0.6

0.8

1

125 250 500 1000 2000 4000

frequency [Hz]

ab

sorp

tio

n c

oeff

icie

nt

0

0.2

0.4

0.6

0.8

1

125 250 500 1000 2000 4000

frequency [Hz]

ab

sorp

tio

n c

oeff

icie

nt

window heavy urtain

0

0.2

0.4

0.6

0.8

1

125 250 500 1000 2000 4000

frequency [Hz]

ab

sorp

tio

n c

oeff

icie

nt

0

0.2

0.4

0.6

0.8

1

125 250 500 1000 2000 4000

frequency [Hz]

ab

sorp

tio

n c

oeff

icie

nt

egg arton glass ber panel, thi kness 50 mm

0

0.2

0.4

0.6

0.8

1

125 250 500 1000 2000 4000

frequency [Hz]

ab

sorp

tio

n c

oeff

icie

nt

0

0.2

0.4

0.6

0.8

1

125 250 500 1000 2000 4000

frequency [Hz]

ab

sorp

tio

n c

oeff

icie

nt

104

panel resonator, 4 mm wood, 120 mm air

layer

audien e on upholstered hair

0

0.2

0.4

0.6

0.8

1

125 250 500 1000 2000 4000

frequency [Hz]

ab

sorp

tio

n c

oeff

icie

nt

0

0.2

0.4

0.6

0.8

1

125 250 500 1000 2000 4000

frequency [Hz]

ab

sorp

tio

n c

oeff

icie

nt

6.6 Cover for porous absorbers

In most ases porous absorbers need a over for me hani al prote tion. Often used are panels with slits

or holes. The openings have to be designed in su h a way that the degree of transmission is lose to

1 in the frequen y range of interest. The problem lies in the high frequen ies

17

. The sound wave an

pass the panel only by an os illation of air olumns in the holes. Due to the inertia this gets more and

more di ult for in reasing frequen y. Figure 6.4 shows the fundamental frequen y dependen y of the

degree of transmission.

0

0.2

0.4

0.6

0.8

1

0.125

0.25

0.5 1 2 4 8

deg

. of

tran

smis

sio

n

normalized frequenzy

Figure 6.4: Normalized frequen y dependen y of the degree of transmission for perforated panels.

The relevant parameters of the panel are the ratio ǫ of the area of the holes relative to the area of the

panel, the diameter r of the holes and the thi kness l of the panel. The length of the os illating air

olumns does not exa tly orrespond to the thi kness of the panel but is a little larger. This fa t is

a ounted for by introdu ing a orre tion 2∆l, resulting in an ee tive panel thi kness of l∗ with

l∗ = l + 2∆l (6.24)

The frequen y f0.5 where the degree of transmission has dropped to 0.5 an be estimated as

f0.5 ≈ 1500ǫ

l∗(6.25)

where

ǫ: ratio of the area of the holes relative to the area of the panel in %

l∗: ee tive panel thi kness in mm

Table 6.1 shows some parameter ombinations for f0.5 = 6300 Hz.

In some ases it may be interesting to expli itly limit the absorption of porous materials at high fre-

quen ies due to the fa t that there is often plenty of high frequen y absorption existent. This an be

done by a proper adjustment of the perforated panel parameters.

17

Fasold, Sonntag, Winkler, Bau- und Raumakustik, Verlagsgesells haft Rudolf Müller, Köln-Braunsfeld, 1987.

105

panel thi kness l 1 mm 1 mm 4 mm 4 mm

ǫ 5 % 10 % 17 % 20%

hole diameter r 0.5 mm 3 mm 0.5 mm 3 mm

Table 6.1: Parameters of a perforated panel for f0.5 = 6300 Hz.

106

Chapter 7

Room a ousti s

7.1 Introdu tion

What makes sound elds in rooms spe ial is the superposition of dire t sound and many rst and

higher order ree tions. As a onsequen e of the sound energy that is stored in these ree tions, there

is so alled reverberation. After swit hing o a sour e in a room, the sound pressure is only slowly

fading away.

This reverberation ee t is obje tively des ribed by the parameter reverberation time T . For a

thorough dis ussion of reverberation, see e.g. Blesser

1

.

From an a ousti al point of view the limiting surfa es (walls, oor and eiling) are the relevant elements

of a room. The sound eld is inuen ed by their geometry, their absorption properties and their

diusivity. For the investigation of the sound eld three methods are in use

• Statisti al room a ousti s assumes a diuse sound eld as a entral simpli ation. The analysis

fo uses on the ratio of dire t and diuse sound and deals with the reverberation. Walls, oor and

eiling are des ribed by the statisti al absorption oe ient αs.

• Geometri al room a ousti s models the sound propagation as energy that propagates along

straight sound rays. This is a high frequen y approximation that holds for wave lengths that

are mu h smaller than the dimensions of the elements of the room. The ree tion properties are

dened by an absorption oe ient and a diusivity to des ribe the s attering behavior.

• Wave based room a ousti s is seeking solutions of the wave equation. The sound propagation

is modeled physi ally orre t and onsiders wave phenomenons su h as resonan e, interferen e

and dira tion. However analyti al solutions are available for a few simple geometries only. In

general, spe i solutions have to be found with numeri al approximations su h as the Boundary

Element method (BEM) or Finite Element method (FEM). The orresponding omputational

eorts restri ts the appli ation to small geometries or low frequen ies. The boundary surfa es have

to be des ribed with their proper impedan es. A di ulty arises as in pra ti e this information is

usually not available.

7.2 Room a ousti s of large rooms

Sound elds in large rooms are hara terized by a high density of room resonan es already at relative

low frequen ies. As a onsequen e the u tuations in the transfer fun tions from a sour e to a re eiver

have arbitrary hara ter. Under these onditions statisti al and geometri al room a ousti methods an

be applied.

7.2.1 Statisti al room a ousti s

Statisti al room a ousti s is based on the on ept of a diuse sound eld, whi h means that

1

Barry Blesser, An Interdis iplinary Synthesis of Reverberation Viewpoints, Journal of the Audio Engineering So iety,

vol. 49, p.867-903 (2001).

107

1. the sound energy density in the whole room is onstant.

2. there is no predominant sound in ident dire tion.

These two onditions are never totally fullled in real situations. However for pra ti al appli ations a

diuse sound eld an be assumed if there is not too mu h absorption in a room and if this absorption

is more or less evenly distributed over the surfa e of the room

2

.

Intensity on a wall

For a given sound energy density w in a room, the sound intensity on a wall shall be determined. The

intensity orresponds to the in oming power per unit area. The power is given by the energy that hits

the onsidered surfa e element dS within one se ond (Figure 7.1).

φ

dS

dV

θ

r

Figure 7.1: Situation to determine the energy ontribution of a volume element dV to the surfa e

element dS in a diuse sound eld.

The energy E that stems from the volume element dV and hits the surfa e element dS is

E =dS cos θ

4πr2wdV (7.1)

In spheri al oordinates the volume dV is

dV = r2drdθ sin(θ)dφ (7.2)

The sound power W , that hits dS within one se ond orresponds to the energy ontribution stemming

from a half sphere with radius R = c× 1 se :

W = IdS =wdS

4π

c×1sec∫

0

2π∫

0

π/2∫

0

cos(θ) sin(θ)dφdθdr =wc

4dS (7.3)

With this the intensity on a wall in a diuse eld with energy density w is found as

I =wc

4(7.4)

Total absorption and power balan e in the diuse eld

If a sound sour e in a room is swit hed on, the sound energy density steadily in reases until a nal state

of sound power balan e is rea hed. This state is hara terized by the ondition that sound power that

is absorbed is just as large as the sound power that is fed to the room by the sour e. The absorption

of the room is des ribed by the total absorption A, dened as

A =

n∑

i=1

Siαi where

n∑

i=1

Si = area of the surfa e of the room (7.5)

2

Murray Hodgson, When is Diuse-Field Theory Appli able? Applied A ousti s, vol.49, n.3, p.197-207 (1996).

108

where αi is the absorption oe ient and Si the area of the room surfa e element with index i, n is

the total number of surfa e elements.

The total sound power that is absorbed by the room surfa e is

Wabsor = IwallA =wc

4A (7.6)

The balan e ondition is

Wabsor =Wsource (7.7)

w =4Wsource

Ac(7.8)

The diuse sound eld an be understood as superposition of many plane waves that arrive from all

possible dire tions. In ase of a plane wave the energy that ows through an area of 1 m

2in 1 se

orresponds to the energy ontained in a ylinder of base 1 m

2and height c× 1 se . With this follows

I = wc =p2

ρc(7.9)

Finally the sound pressure pdiffuse in a diuse eld an expressed in dependen y of the sour e power

Wsource and the total absorption A as

p2diffuse =4Wsourceρc

A(7.10)

Relation 7.10 is valid only under the idealized assumption that the diuse eld is onstant throughout

the room. However there are empiri al formulas to onsider a distan e dependen y of the sound pressure

3

:

p2diffuse =4Wsourceρc

Ae−(

2∂rc )

(7.11)

where

r: sour e - re eiver distan e∂: de ay onstant = 3 ln(10)/TT : reverberation time

c: speed of sound

Dire t sound and diuse eld ontribution, riti al distan e

Up to now only the diuse eld was onsidered. Of ourse a diuse eld an't exist without a dire t

sound eld. Under the assumption of an omnidire tional sour e that ex ites the sound elds, the

pressure square pdirect of the dire t sound is given as:

p2direct =Wsourceρc

4πr2(7.12)

and hen e the total sound pressure square p2 sums up to

p2 = p2direct + p2diffuse =Wsourceρc

(

1

4πr2+

4

A

)

(7.13)

For small distan es r, the rst term in the bra kets dominates. This indi ates that the dire t sound with

its 1/r2 distan e dependen y is larger than the diuse sound. For in reasing distan es the signi an e

of the dire t sound de reases and the lo ation independent diuse sound eld determines more and

more the total sound pressure (Figure 7.2). The distan e where dire t and diuse sound have equal

strengths is alled riti al distan e and is usually labeled as rc.

with

1

4πr2c=

4

Afollows rc =

√

A

16π(7.14)

109

1 10

distance to source [m]

40

45

50

55

60

65

70

sound p

ressure

level [d

B]

direct

diffuse

total

Figure 7.2: Distan e dependen y of sound pressure in a room with dire t sound and an ideal diuse

sound eld. The arrow marks the riti al distan e where dire t and diuse sound have equal strength.

If the sour e shows enhan ed radiation in one dire tion, the riti al distan e in this dire tion in reases

a ordingly.

In reality the distan e dependen y of sound pressure in a room doesn't follow exa tly the relation shown

in Fig. 7.2. A more subtle des ription is based on Eq. 7.11 and yields

4

:

L(r) = 10 log

(

100

r2+

31200Te−0.04r/T

V

)

[dB (7.15)

where

L(r): sound pressure level at distan e r relative to the value in 10 m

T : reverberation time [se

V : room volume [m

3

Figure 7.3 shows the orresponding distan e dependen y of sound pressure for a room with volume V= 20'000 m

3and a reverberation time T = 2 se .

2.9 3.4 4.1 4.8 5.7 6.8 8.1 9.7 11.5 13.7 16.2 19.3 23.0 27.3 32.5 38.6

distance [m]

−15

−10

−5

0

5

10

15

sound p

ressure

level [d

B]

direct

direct+diffuse

Barron

Figure 7.3: Sound pressure level as a fun tion of distan e in a room with V = 20'000 m

3and a

reverberation time T = 2 se . Dire t sound, diuse eld theory and the formula of Barron are shown.

3

M. Barron, L. J. Lee, Energy relations in on ert auditoriums, Journal of the A ousti al So iety of Ameri a, vol 84,

p.618-628 (1988).

4

M. Barron, Loudness in Con ert Halls, A usti a - a ta a usti a, vol.82, suppl. 1 (1996).

110

Reverberation, reverberation time

Above, the sound power relations for the stationary ondition have been dis ussed. In the following, the

situation of a sound sour e that is swit hed o shall be investigated. Due to the energy that is stored

in the ree tions, the sound energy density in the room de reases only slowly, depending on the room

volume and the absorption of the room surfa es. This pro ess is alled reverberation and des ribed

quantitatively by the so alled reverberation time. For the power balan e an be written

Wsource =Wabsor + Vdw

dt(7.16)

where

Wsource: sound power emitted by the sour e

Wabsor: sound power that is absorbed by the room surfa es

V : room volume

w: energy density

From

Wabsor =wc

4A (7.17)

follows

Wsource =wc

4A+ V

dw

dt(7.18)

Eq. 7.18 represents a dierential equation for the energy density w. If the sour e is swit hed o, the

reverberation pro ess manifests. The solution of w(t) that fullls the equation

0 =wc

4A+ V

dw

dt(7.19)

has the form

w(t) = w0ebt

(7.20)

where

b = − cA4V

Eq. 7.20 des ribes the reverberation pro ess as an exponentially de aying time history. This orresponds

to a straight line in the level-time representation as shown in 7.4.

0 0.5 1

time [s]

−40

−30

−20

−10

0

10

sound p

ressure

level [d

B]

Figure 7.4: Example of sound pressure de ay in a room after swit hing o the sour e at time t = 0.

The reverberation time T is dened as the time that passes until the energy density has de reased to

1E-6 of its initial value. In the dB s ale this orresponds to -60 dB. From

111

e−cA4VT = 10−6

(7.21)

follows

T =− ln(10−6)4V

cA=

0.16V

A(7.22)

The relation between T , the room volume V and total absorption A was found experimentally by W.

C. Sabine in 1900. To his honor, Eq. 7.22 is usually alled Sabine equation.

An other derivation of the reverberation time was given by Eyring. His on eption was that sound

propagates in form of energy pa kets along straight lines. Whenever su h a pa ket hits a room surfa e,

a ertain amount of energy is absorbed while the remaining energy is ree ted. Besides the average

absorption oe ient α of the room surfa es, the mean free path length between two ree tions ℓ isthe se ond relevant parameter. For a re tangular room ℓ an be al ulated from the volume V and the

room surfa e area S:

ℓ =4V

S(7.23)

The reverberation pro ess an now be observed for one single energy pa ket. It is assumed that the

average absorption oe ient over the whole room surfa e is α with (α = 1/S∑

Siαi). At ea h

ree tion the energy is redu ed by α×100 %. Thus after N ree tions the remaining energy E is

E(N) = E0(1− α)N (7.24)

The de ay to 1E-6 of the initial energy is rea hed after M ree tions where

M =−13.8

ln(1− α)(7.25)

M ree tions orrespond to a path length L =M · ℓ, or a time T

T =Mℓ

c=

−13.8× 4V

ln(1− α)cS=

0.16V

− ln(1− α)S(7.26)

For little absorption (α → 0) the reverberation formula of Eyring (Eq. 7.26) approximates the formula

of Sabine (Eq. 7.22). For highly damped rooms (α → 1) the formula of Eyring takes on the reasonable

value T = 0, while Sabines formula predi ts a value T > 0. Eyring predi ts in any ase a lower

reverberation time than Sabine.

At high frequen ies, air absorption may be ome a relevant fa tor that inuen es reverberation. This

an be onsidered by introdu ing an additional fa tor in the Eyring reverberation formula:

E(N) = E0(1− α)Ne−mNl′

(7.27)

where

m : intensity damping onstant for air a ording to Table 7.1

The above derivation (Eq. 7.26) for the reverberation time T is a ordingly modied with Eq. 7.27 as

T =0.16V

− ln(1− α)S + 4mV(7.28)

For outdoor sound propagation appli ations, omprehensive tables of air absorption oe ients are

available (ISO 9613-1). The air absorption is spe ied by a oe ient α that des ribes the level

redu tion in dB per meter. The damping onstant m used here an be expressed in α as

m = ln(100.1·α) (7.29)

112

relative humidity [% 500 Hz 1000 Hz 2000 Hz 4000 Hz 8000 Hz

30 0.00058 0.00115 0.00325 0.01125 0.03874

40 0.00060 0.00107 0.00258 0.00838 0.02992

50 0.00063 0.00107 0.00228 0.00683 0.02423

60 0.00064 0.00111 0.00214 0.00590 0.02047

70 0.00064 0.00115 0.00208 0.00531 0.01787

80 0.00064 0.00119 0.00207 0.00493 0.01599

Table 7.1: Intensity damping onstant m of air as a fun tion of frequen y and relative humidity at a

temperature of 20

.

Rooms with non-diuse behavior

Besides the above mentioned ases where a diuse eld establishes and thus the energy density shows

an exponential de ay, there are room situations with a deviating de ay urve. This is the ase for

rooms with very inhomogeneous distribution of the absorption or oupled rooms where two rooms with

dierent damping are arranged that they an ommuni ate with ea h other.

Table 7.2 shows the al ulated reverberation times for a re tangular room with dierent absorber

ongurations and varying degree of diusivity of the surfa es. In any ase the total absorption was

kept onstant. The al ulations were performed with a ray tra ing model (see next se tion).

surfa e diusivity T al ulation a ording to Sabine 1.33 s

ray tra ing, absorption on entrated on one surfa e of 20x15 m 30% 2.10 s

ray tra ing, absorption distributed on the whole surfa e 30% 1.32 s

ray tra ing, absorption on entrated on one surfa e of 20x15 m 90% 1.07 s

Table 7.2: Comparison of al ulated reverberation times T in a re tangular room with dimensions 20

x 15 x 6.7 m = 2000 m

3and total absorption of 240 m

2for dierent distributions of the absorbing

surfa es. The olumn surfa e diusivity des ribes the assumed diusivity of the ree ting surfa es.

For equally distributed (homogeneous) absorption the ray tra ing al ulation is very lose to the Sabine

result. However for on entrated absorption and low diusivity the reverberation times an in rease

onsiderably.

A typi al example of oupled rooms is a hall with a foyer that gets sound energy from the hall by doors

or other small openings. Further examples are hur hes with adja ent hapels. If the sour e is lo ated

in the room with less absorption, a de ay urve as shown in Fig. 7.5 will o ur.

Absorption of audien e

In many rooms, espe ially in on ert halls, the audien e ontributes signi antly or even dominates the

absorption. It is therefore of great importan e to know the orresponding absorption hara teristi s

pre isely. However the exa t absorption oe ient depends on dierent fa tors su h as density and

arrangement of the seating, the upholstering of the seats or the type of lothes people are wearing.

Typi al α values are given in Table 7.3

5

.

125 Hz 250 Hz 500 Hz 1000 Hz 2000 Hz 4000 Hz

upholstered seat, row spa ing 1.15 m 0.30 0.35 0.50 0.60 0.70 0.70

Table 7.3: Typi al absorption oe ients αs for audien e areas.

5

Fasold, Sonntag, Winkler, Bau- und Raumakustik, Rudolf Müller Verlag, 1987.

113

-30

-20

-10

0

10

0 200 400 600 800 1000

time [ms]

sou

nd

pre

ssu

re leve

l [d

B]

Figure 7.5: Sound de ay for two oupled rooms where the sour e is in the room with lower absorption

and the re eiver in the room with higher absorption.

Statisti al impulse responses

For general, non-spe i room a ousti al investigations statisti al impulse response models may be

of interest. In the ontext of statisti al room a ousti s su h a model has to dene the dire t sound

and the diuse eld ontribution. Thus the ne essary spe ifying parameters are sour e dire tivity,

distan e between sour e and re eiver, room volume and absorption. The dire t sound is represented as

a Dira pulse with appropriate amplitude and delay. The diuse eld ontribution is simulated by an

exponentially de aying noise signal. With this a statisti al, time dis rete impulse response h(i) an be

written as

6

:

h(i) =

√Γ

r∆

(

i− trun

(

fcr

c

))

+

√

4πc

V fce−

i∂fc ξ(i)θ

(

i − trun

(

fcr

c

))

(7.30)

where

i: sample number

Γ: dire tivity fa tor as ratio of the intensity in dire tion of the re eiver and the intensity averaged over

all dire tions

r: sour e - re eiver distan e∆(i): impulse fun tion, = 1 for i = 1, elsewhere 0

trun (): trun ate-fun tion, round o to the next lower whole number

fc: lo k frequen y

c: speed of sound

V : room volume

∂: de ay onstant of the room, ∂ = 3 ln(10)/T (T : reverberation time)

ξ(i): sequen e of samples of white noise, ξ(i) and ξ(i + 1) are independent samples of a normally

distributed random variable with mean = 0 and standard deviation = 1

7

θ(i): step fun tion, = 1 for i ≥ 1, elsewhere 0

Figure 7.6 shows a statisti al impulse response that was reated with the above pro edure.

7.2.2 Geometri al room a ousti s

Geometri al a ousti s assumes that sound propagates in form of rays along straight lines. This geo-

metri al approa h is a high frequen y approximation and ignores wave phenomena su h as dira tion

6

U. P. Svensson, Energy-time relations in a room with an ele troa ousti system, Journal of the A ousti al So iety of

Ameri a, vol. 104, p.1483-1490 (1998).

7

Normally distributed random numbers an be generated from equally distributed random numbers as follows: get

two random numbers RA and RB that are equally distributed in the interval (0,1), then onvert them to two normally

distributed random numbers SA and SB with standard deviation σ a ording to:

SA = σ√

−2 ln(RA) cos(2πRB)

SB = σ√

−2 ln(RA) sin(2πRB)

114

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.1 0.2 0.3 0.4 0.5 0.6

time [s]

Figure 7.6: Example of an arti ially generated room impulse response with a de ay onstant ∂ = 6.9( orresponding to a reverberation time T = 1 se ), a room volume V = 10'000 m

3and a sour e -

re eiver distan e r = 15 m.

or interferen e.

Ree tion at plane surfa es, spe ular sour es

If a sound ray hits a surfa e, it looses a ertain amount of its energy depending on the absorption

oe ient of the orresponding surfa e. The remaining energy is ree ted a ording to the law of

ree tion (angle of in iden e = angle of ree tion). A ertain sound path an be determined by

onstru tion of mirror sour es (see Fig. 7.7).

receiver

source

Q'

Q''

Figure 7.7: Constru tion of the ree tion of sound rays by introdu tion of mirror sour es.

Ree tion at stru tured surfa es, diuse ree tion and s attering

A ree tion at a surfa e with signi ant depth stru turing is no longer spe ular but rather diuse like.

The degree of diusivity depends on the ratio of the stru ture dimension and the wave length. Diuse

ree tions usually o ur at higher frequen ies while low frequen ies show spe ular behavior. More

spe i ally, three ases an be distinguished as shown in Fig. 7.8.

A diuse ree tion returns sound energy into a large solid angle. Often the idealized Lambert ree tion

hara teristi s is assumed

8

. It states that the intensity of the ree tion in dire tion φ relative to the

surfa e normal is proportional to the osine of φ.

Energy impulse response

Within the on ept of geometri al room a ousti s, sound propagation is modeled by aid of energy

pa kages that travel along straight lines (sound rays). After emission at the sour e the pa kages that

8

Max Born, Emil Wolf, Prin iples of Opti s, Pergamon Press, 1980.

115

s

s

s

Figure 7.8: Ree tion at a stru tured surfa e. Top: For λ ≫ stru ture dimension s → the stru ture

has no ee t → spe ular ree tion at an 'average' plane. Middle: For λ ≈ stru ture dimension s →the stru ture a ts as a whole → diuse ree tion. Bottom: λ ≪ stru ture dimension s → the single

stru ture elements a t as ree tors → spe ular ree tion at the stru ture details.

φIo

Figure 7.9: Ideal diuse ree tion a ording to Lambert. Independent of the sound in iden e dire tion

the intensity of the ree tion in dire tion φ is proportional to cos(φ).

arrive at a re eiver an be olle ted and registered with regard to the energy they represent and their

travel time. This olle tion orresponds to an energy impulse response (Fig. 7.10) for the hosen sour e

and re eiver position.

Obje tive room a ousti al riteria

For the onsidered sour e and re eiver position the energy impulse response represents the nger print

of the room. In the past, many dierent features of su h impulse responses have been proposed to

relate the subje tive quality of a room to obje tive riteria. From the large atalogue, a small set of

these riteria has proven to be su ient and relevant to des ribe the a ousti al quality of rooms

9

.

These riteria are usually evaluated for the o tave bands from 125 Hz to 4 kHz. In the following, the

origin of the time axis t = 0 is understood as the moment of arrival of the dire t sound.

• Reverberation time T [s

The reverberation time is the most fundamental feature to des ribe the room a ousti al properties.

It has global hara ter, whi h means that the value is not hanging a lot for dierent positions.

The reverberation time is usually measured with ba kward integration of the squared impulse

response. The de ay urve is then evaluated between -5 and -35 dB. This time is doubled to get

the reverberation de ay of 60 dB.

9

ISO Norm 3382 Measurement of the reverberation time of rooms with referen e to other a ousti al parameters. 1997.

116

Projekt: KIZA2.GEO MatVarNr: 1 ResFileNr: 3 Frequenz: 1000

0.000 0.250

Figure 7.10: Example of an energy impulse response. The earliest ontribution orresponds to the dire t

sound. Then rst and higher order ree tions follow with in reasing density. Note the unusual strong

ree tion due to fo using ee ts of a on ave room surfa e.

• Early De ay Time EDT [s

The Early De ay Time EDT is dened similarly to the reverberation time, but is based on the

de ay over the top 10 dB. This time is then multiplied by 6 to extrapolate for a de ay over 60 dB.

From a subje tive point of view the EDT is more relevant for a listener, as the dynami range

for musi performan es is typi ally in the order of 10. . .20 dB. The EDT may depend strongly

on the listening position. The just audible dieren e of a variation of EDT is in the order of 5

% in an A/B omparison

10

.

• Clarity C80 [dB

Clarity measures the ratio of early arriving energy relative to the late energy in the impulse

response. C80 des ribes the transparen y of musi . With the energy impulse response h2(t), larity is al ulated as follows:

C80 = 10 log

80ms∫

0

h2(t)dt

∞∫

80ms

h2(t)dt

(7.31)

A typi al value for C80 is 0 dB, an in rease of the value means higher larity. The just audible

dieren e is in the order of 0.5 dB in the dire t A/B omparison.

• Strength G [dB

The strength G is a measure that des ribes the level at the re eiver position relative to the level

under free eld onditions at a distan e of 10 m. If the sour e re eiver distan e is 10 m, Gspe ies dire tly the ampli ation by the room. The strength is found by integration over the

energy impulse response h2(t):

G = 10 log

∞∫

0

h2(t)dt

∞∫

0

h2f,10m(t)dt

(7.32)

where

hf,10m: energy impulse response under free eld onditions at 10 m distan e.

The just audible dieren e is about 1 dB in a dire t A/B omparison.

• Deutli hkeit D50 [%

Similarly to larity C80, Deutli hkeit D50 des ribes the learness of a room a ousti al situation.

D50 is dened as the energy ratio of useful early energy up to 50 ms after the dire t sound relative

to the total energy in the impulse response. D50 is mainly used to investigate the learness of

spee h signals. With the energy impulse response h2(t) D50 is found as

10

M. Vorländer, International Round Robin Test on Room A ousti al Computer Simulation, ICA 1994 Bergen.

117

D50 =

50ms∫

0

h2(t)dt

∞∫

0

h2(t)dt

× 100% (7.33)

A D50 value of 40 % orresponds to an intelligibility of syllables of about 87 %, a D50 of 60 %

means an intelligibility of syllables of about 93 %. The just audible dieren e is about 5 % in the

dire t A/B omparison.

• Center time TS [ms

The enter time des ribes similarly to C80 and D50 the temporal distribution of in oming energy.

However TS avoids stri t separations to distinguish between bene ial and detrimental energy.

TS orresponds to the enter of gravity of the energy impulse response h2(t):

TS =

∞∫

0

th2(t)dt

∞∫

0

h2(t)dt

(7.34)

The just audible dieren e is about 10 ms in the dire t A/B omparison.

• Lateral energy fra tion LF [%

The lateral energy fra tion measures the ratio of early lateral energy relative to early omnidire -

tional energy. The LF des ribes spa iousness whi h is a result of inter-aural signal dieren es.

The evaluate LF the energy impulse response has to be determined on e with an omnidire -

tional mi rophone ( → h2(t)) and on e with a gure of eight mi rophone ( → h2∞(t)) where theorientation has be hosen in su h a way that the sensitivity in frontal dire tion is zero.

LF =

80ms∫

0

h2∞(t)dt

80ms∫

0

h2(t)dt

× 100 (7.35)

The just audible dieren e is about 5 % in the dire t A/B omparison.

For reverberation times T there is onsensus about optimal values as a fun tion of room volume for

a wide variety of dierent appli ations. Fig. 7.11 shows optimal values in the mid frequen y range

for musi and spee h performan e. In general one aims at reverberation times that are more or less

independent of frequen y. In on ert halls however a slight in rease at lower frequen ies is usually

per eived as bene ial (warmer sound).

For the other obje tive riteria, only preliminary optimal values exist due to la k of su ient experien e.

For onvert halls the values in Table 7.4

11

may be applied.

parameter EDT C80 (500. . .2 kHz) G (500. . .2 kHz) LF (125. . .1 kHz)

optimal range 1.8. . .2.2 s -2. . .+2 dB > 0 dB 0.1. . .0.35

Table 7.4: Values of further room a ousti al riteria onsidered as optimal in on ert halls.

7.2.3 A ousti al design riteria for rooms

The design of a room for good room a ousti s has to onsider dierent aspe ts that vary in their

relevan e depending on the fun tion and the usage. The most important riteria are:

Silen e Any audible noise that has nothing to do with the performan e on stage has to be avoided.

Possible unwanted noise in auditoriums may stem from external tra or from adja ent rooms.

An other possible noise sour e is the air onditioning system of the auditorium.

11

M. Barron, Auditorium A ousti s and Ar hite tural Design. 1993

118

100 316 1000 3162 10000 31623 100000

room volume [m3]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

reve

rbera

tio

n t

ime [

s]

speech

music

Figure 7.11: Optimal values of the reverberation time at mid frequen ies in dependen y of the room

volume for spee h and musi performan e.

Dire t sound The whole audien e area should re eive su ient dire t sound from the sour e. Early

ree tions (within 50 ms) an support the dire t sound supply.

Reverberation Depending on usage, room volume and room type, an appropriate reverberation time

has to be adjusted.

Lateral ree tions The feeling of spa iousness is triggered by un orrelated signals at the two ears of

a listener. This makes strong lateral ree tions ne essary.

Diusivity With the ex eption of early lateral ree tions, the ree tions should typi ally be diuse and

not spe ular. This spreads ree ted sound energy over time and redu es the danger of fo using

ee ts.

Balan e Dierent se tions of extended sour es su h as or hestras should be heard in the audien e with

equal strength.

Audibility on stage To guarantee an optimal performan e, the musi ians in an or hestra should hear

ea h other reasonably well.

For ertain room types or usages, spe i re ommendations exists regarding the a ousti al design:

• rooms for spee h ommuni ation up to a room volume of about 5'000 m

3su h as onferen e

rooms, s hools or restaurants

12

.

• re ording studios

13

.

7.2.4 Room a ousti al design tools

The optimal a ousti al design of a room requires appropriate analysis tools. They help to proof the

e a y of planned measures. Depending on the questions asked, a variety of design tools are available.

Constru tion of sound rays

A preliminary estimate of the sound distribution in a room an be a hieved by the onstru tion of

sound rays by hand. Thereby one usually restri ts to a horizontal or verti al se tion through the room.

Assuming an omnidire tional sour e some ten or twenty sound rays are drawn in all dire tions. At

12

Hörsamkeit in kleinen bis mittelgrossen Räumen, DIN 18041.

13

DIN 15996, Elektronis he Laufbild- und Tonbearbeitung in Film-, Video- und Rundfunkbetrieben (1996).

119

the interse tions with boundary surfa es the rays are ree ted. The resulting density of the rays at a

spe i re eiver lo ations determines the sound pressure level at that point. The manual onstru tion

of sound rays is suitable for example to investigate fundamental eiling shapes or the optimal orientation

of ree tors. The eort to onstru t higher order ree tions grows qui kly, one will then use ray tra ing

omputer models.

Cal ulation of reverberation times

As mentioned above, the reverberation time is the most fundamental room a ousti al parameter. If

the materialization is known, the reverberation time of a room an be al ulated by appli ation of the

Sabine or Eyring formula. In on ert halls, the audien e is usually the dominating absorber. In these

ases it is therefore possible to estimate the reverberation time T with the area of the audien e SP ,the room volume V as:

T ≈ 0.15V

SP(7.36)

S ale models

Sound propagation in rooms an be simulated with help of s ale models

14

,

15

,

16

,

17

. If all dimensions

are s aled by a fa tor 1/s and at the same time the frequen y is s aled by s (preservation of the ratio of

wavelength and dimension) the sound propagation phenomena remain unaltered. A di ulty is to nd

materials for the s ale models that have similar absorption hara teristi s in the transformed frequen y

domain as the original material in the original frequen y domain. In addition, strategies are ne essary

to over ome the strong air absorption in the s ale model frequen y range (up to 50 kHz). One solution

is to dry the air down to a relative humidity of a few per ent- Under these onditions the air shows low

absorption up to high frequen ies. An other approa h is to ompensate for the absorption by way of a

al ulation. As travel times have to be known this an only be done on basis of the impulse response.

Typi al values for the s ale fa tor s are between 10 and 50.

Computer simulations

Nowadays it be omes more and more ommon to use omputer software to simulate sound propagation

in rooms. The rst attempt in this dire tion was most probably made by S hroeder

18

, however the

rst who a tually wrote a omputer program were Krokstad and his olleagues

19

.

Room a ousti al omputer simulations an be divided roughly into two ategories. The rst ategory

omprises numeri al methods that nd solutions to the wave equation. The se ond ategory ontains

methods that simulate sound propagation based on geometri al a ousti s.

All numeri al methods that solve the wave equation have in ommon that the room volume and/or

the room surfa e have to be dis retized. The orresponding mesh has to be signi antly ner than

the shortest wave length of interest. The omputational eort be omes extremely high for large rooms

and high frequen ies.

The methods based on geometri al a ousti s assume sound propagation along straight lines. Wave

phenomena su h as interferen e or resonan e an not be onsidered. Computer models based on

geometri al a ousti s an be split into two groups: ray tra ing and mirror sour es.

14

F. Spandö k, Akustis he Modellversu he, Annalen der Physik, vol. 20, 1934, p.345.

15

A. F. B. Ni kson, R. W. Mun ey, Some experiments in a room and its a ousti model; A usti a, 1956, vol. 6,

p.295-302

16

D. Brebe k, R. Bue klein, E. Krauth, F. Spandö k, Akustis h ähnli he Modelle als Hilfsmittel für die Raumakustik,

A usti a, 1967, v.18, p.213-226.

17

J. D. Pola k, A. H. Marshall, G. Dodd, Digital evaluation of the a ousti s of small models: The MIDAS pa kage,

Journal of the A ousti al So iety of Ameri a, 1989, v.85, p.185-193.

18

M. R. S hroeder, B. S. Atal, C. Bird, Digital Computers in room a ousti s, Pro . 4th Intern. Congr. of A ousti s,

1962, Paper M21.

19

A. Krokstad, S. Strom, S. Sorsdal, Cal ulating the a ousti al room response by the use of a ray tra ing te hnique,

Journal of Sound and Vibration, 1986, p.118-124.

120

Ray tra ing methods

20

simulate sound propagation by emitting many sound parti les at the sour e

position (Fig. 7.12). The parti les propagate along straight lines. If a parti le hits a boundary surfa e,

the energy is redu ed orresponding to the absorption oe ient of the surfa e. The parti le with

adjusted energy is then ree ted based on a ertain ree tion hara teristi s that is des ribed by a

diusivity fa tor. If a parti le is ree ted diusely, the outgoing dire tion φ is determined randomly

where the probability of a ertain angle φ is proportional to osine of φ. At ea h re eiver position a

sphere of small diameter is onstru ted. Ea h time a sound parti le passes su h a re eiver volume, the

orresponding travel time and energy of the parti le is noted in a table.

With the mirror sour e method, all possible sound paths between a sour e and a re eiver are determined

by onstru ting all visible mirror sour es up to a ertain order

21

. All room surfa es are assumed to

ree t spe ularly. The attenuation of a ertain sound path is given by the produ t of the absorption

oe ients of all surfa es involved and a fa tor 1/d2 with d the travel distan e.

The ray tra ing or mirror sour e method deliver nally an energy impulse response for the room and

the hosen sour e and re eiver points. From this the above mentioned room a ousti al riteria su h

as EDT or C80 an be evaluated. Furthermore sound pressure impulse responses an be derived for

auralization purposes. For a re ent overview of geometri al room a ousti modeling see the tutorial

paper by Savioja

22

.

Figure 7.12: Example of the beginning of a ray tra ing simulation.

Auralization

As seen above there are dierent parameters to evaluate and des ribe the a ousti al quality of

a room. These parameters an be al ulated in advan e during the planning phase of a proje t.

However the ultimate riterion is the listening experien e in the room. The pro ess of simulating the

audible impression of a room is alled auralization. First attempts of auralization with help of s ale

models go ba k to Spandö k

23

,

24

. Thereby the signal of interest was up-shifted in frequen y by

an appropriate s ale fa tor and emitted in the s ale model. At the listener position the signal was

re orded, down-shifted in frequen y and played ba k through headphones.

With the introdu tion of room a ousti al omputer simulations, a new auralization approa h was in-

trodu ed

25

,

26

,

27

. With help of the omputer simulation it is determined, when how mu h energy from

20

M. Vorlaender, Ein Strahlverfolgungsverfahren zur Bere hnung von S hallfeldern in Raeumen, A usti a, 1988, v.65,

p.138-148.

21

J. B. Allen, D. A. Berkley, Image method for e iently small-room a ousti s, Journal of the A ousti al So iety of

Ameri a, 1979, v.65, p.943-950.

22

L. Savioja, U. P. Svensson, Overview of geometri al room a ousti modeling te hniques, J. A oust. So . Am. 2015,

vol. 138, 708-730.

23

F. Spandö k, Annahlen der Physik V, vol. 20, 1934, p.345

24

F. Spandö k Das Raumakustis he Modellverfahren mit massstabsgere hter Frequenztransponierung und die

Mögli hkeiten seiner Verwirkli hung. Third International Congress of A ousti s, 1959, pp. 925-928.

25

Hilmar Lehnert, Jens Blauert, Prin iples of Binaural Room Simulation, Applied A ousti s, 1992, v.36, p.259-291.

26

Mendel Kleiner, Bengt-Inge Dalenbae k, Peter Svensson, Auralization - An Overview; Journal of the Audio Engineering

So iety, 1993, v.41, p.861-875.

27

L. Savioja, et. al. Creating Intera tive Virtual A ousti Environments, Journal of the Audio Engineering So iety, vol.

121

whi h dire tion hits the re eiver. A ording to this distribution, the signal of interest is then delayed

a ordingly and played ba k over a loud of loudspeakers installed in an ane hoi hamber

28

(Figure

7.13).

Figure 7.13: S hemati representation of a loud of loudspeakers distributed around a listener position

to auralize the a ousti s of an auditorium. The loudspeakers are fed with appropriate delayed and

weighted opies of the reverberation free sour e signal.

A serious drawba k of the loudspeaker loud is the spa e requirements and the need for an ane hoi

room. Indeed all that has to be done with auralization is to produ e appropriate signals at the two

eardrums of the listener. It should therefore be possible to realize an auralization playba k system with

help of headphones

29

. To do so, additional information about the head related transfer fun tions

(HRTF) is ne essary. As dis ussed above, the room a ousti al simulation delivers impulse responses

for dierent ategories of in iden e angles. The room impulse responses between sour e and the

two eardrums are obtained by onvolution with the orresponding HRTFs. Finally the headphone

auralization signals are generated as onvolution of the dry sour e signal with the two room impulse

responses to the eardrums.

Compared to the loudspeaker loud solution two problems are asso iated with the auralization by

headphones. The rst di ulty is the fa t that the head related transfer fun tions dier from person

to person. For optimal results these HRTFs should be determined individually. The se ond problem is

that the headphone representation an not map head movements

30

.

Most of today's software pa kages for room a ousti al simulations allow for auralization by headphones.

7.2.5 Some room a ousti al ee ts that are not onsidered with statisti al

or geometri al a ousti s

The modeling of sound propagation in rooms by means of statisti al or geometri al a ousti s ignores

the wave nature of sound and is therefore only a oarse approximation to reality. In the following a few

aspe ts are dis ussed that may have relevan e in rooms but are usually not onsidered.

Sound propagation at grazing in iden e over audien e areas

If sound propagates at grazing in iden e over audien e areas, additional damping an be observed.

This is rstly due to destru tive interferen e between dire t sound and sound that is ree ted and/or

s attered at heads and shoulders of the audien e and se ondly due to energy that is lost as a onsequen e

47, p.675-705 (1999)

28

Y. Korenaga, Y. Ando, A Sound-Field Simulation System and Its Appli ation to a Seat-Sele tion System, J. Audio

Eng. So ., vol. 41, 1993, pp. 920-930.

29

K. H. Kuttru, Auralization of Impulse Responses Modeled on the Basis of Ray-Tra ing Results, J. Audio Eng. So .,

vol. 41, 1993, pp. 876-880.

30

A solution to over ome this di ulty is the implementation of head tra king systems that apture the orientation of

the head and adjust the headphone signals a ordingly

122

of dira tion. This additional damping is alled seat dip ee t in the literature

31

,

32

,

33

,

34

. Figure 7.14

shows measurements of Mommertz

35

demonstrating the order of magnitude of the seat dip ee t.

Figure 7.14: Frequen y response of the additional damping for sound propagating at grazing in iden e

over an audien e area. The measurement position was in the 12th row at a height of 1.2 m. The height

of the sour e varied between 1.2 and 2.0 m.

Ree tion at nite surfa es

The ree tion of sound waves at hard surfa es of innite extension an be handled with the mirror

sour e on ept. This is a fundamental assumption behind geometri al room a ousti al tools. However

this on ept is no longer fully orre t for small ree tors, low frequen ies and grazing sound in iden e.

In these ases where the extension of the ree tor has to be taken into a ount, the on ept of Fresnel

zones may help to identify the frequen y dependent dimension that is ne essary for a full ree tion.

For a given ree tor geometry (Fig. 7.15), the lower limiting frequen y fu for full ree tion an be

estimated with Eq. 7.37

36

fu =2c

(l cosβ)2dQRdRE

(dQR + dRE)(7.37)

where

c: speed of sound [m/s

dQR: distan e sour e → point of ree tion [m

dRE : distan e point of ree tion → re eiver [m

l: dimension of the ree tor [m

β: angle of in iden e relative to the ree tor normal dire tion

31

E. Meyer, H. Kuttru, F. S hulte. Versu he zur S hallausbreitung über Publikum. A usti a, vol.15, 1965, p.175-182.

32

S. Bradley. Some further investigations of the seat dip ee t. J. A ousti al So iety of Ameri a, vol. 90, 1991,

p.324-333.

33

R. He ht, E. Mommertz. Ein S hallteil henverfahren zur Simulation der streifenden S hallausbreitung über Publikum.

DAGA 94, 1994, p.229-232.

34

D. Takahashi, Seat dip ee t: the phenomena and the me hanism, J. A ousti al So iety of Ameri a, vol. 102, 1997,

p.1326-1334.

35

E. Mommertz. Einige Messungen zur streifenden S hallausbreitung über Publikum und Gestühl. A usti a, vol. 79,

1993, p.42-52.

36

ISO 9613-2, A ousti s - Attenuation of sound during propagation outdoors - Part 2.

123

l

β

Q

E

d REd QR

Figure 7.15: Situation for the estimation of the lower limiting frequen y for a full ree tion at a ree tor

of limited size.

7.2.6 Ree tions at spheri al surfa es

Curved stru tures and on ave room shapes need spe ial attention

37

,

38

. Convex urvatures are

unproblemati under normal onditions as they in rease s attering of ree ted sound energy. Con ave

urvatures on the other hand show often unwanted fo using ee ts with highly inhomogeneous sound

eld distributions. Spe ta ular examples are whispering galleries that allow for ommuni ation between

distant points with unnatural low damping. There exist quite a few histori al buildings that ontain

sound fo using elements. From a today's perspe tive it is not lear whether these amplifying ee ts

have been implemented deliberately or whether they are a produ t of a ident

39

.

In many ases domed stru tures an be approximated by parts of a sphere. In two dimensions, this

leads to the dis ussion of ree tion of rays at a small ar of a ir le (Fig. 7.16).

k2

k1

aAB

C

D

E

Figure 7.16: Situation of the ree tion of sound rays at a small ar of a ir le. On the ir le k1 with

enter A the ar between the points C and D is onsidered. The sour e position is assumed on the

line a or on the ir le k2 where the points B and A dene the diameter of k2.

Ree tion at ir les: sour e position on axis a

If the sour e point lo ation is on the axis a (see Fig. 7.16), emitted sound rays are ree ted as shown

in Fig. 7.17.

37

M. Ver ammen, Sound Ree tions from Con ave Spheri al Surfa es. Part I: Wave Field Approximation, A ta A usti a

united with A usti a, vol. 96, 82-91 (2010).

38

M. Ver ammen, Sound Ree tions from Con ave Spheri al Surfa es. Part II: Geometri al A ousti s and Engineering

Approa h, A ta A usti a united with A usti a, vol. 96, 92-101 (2010).

39

K. Heuts hi, Akustik der Evangelis hen Filialkir he in Guarda-Giarsun, S hweizer Ingenieur und Ar hitekt SI+A Nr.

41 (2000).

124

Figure 7.17: Ree tion of sound rays at a ir le for dierent sour e positions.

A ording to Figure 7.17 the ree tion at a ir le an show hyperboli , paraboli or ellipti behavior,

depending on the sour e position in relation to the enter of the ir le. For a mathemati al dis ussion,

a unity ir le is assumed with enter at xZ = 1.0, yZ = 0.0. The ir le is then des ribed by Eq. 7.38

or 7.39.

(x− 1)2 + y2 = 1 (7.38)

or

y2 = 2x− x2 (7.39)

In the following it is assumed that the sour e position is on the x-axis and that sound rays are emitted

into a small angular segment in −x-dire tion. For the ree tion only a small region of the ir umferen e

(x small) is of interest. Eq. 7.39 an then be approximated by

y2 ≈ 2x (7.40)

It an be shown that Eq. 7.40 approximates a small portion of an ellipse, a parabola or a hyperbola.

The behavior of the ree tion an easily be dis ussed if the sour e point is interpreted as the fo al

point of the orresponding oni se tion.

Ellipse The equation for an ellipse as shown in Fig. 7.18 is given by:

(x− a)2

a2+y2

b2= 1 (7.41)

where:

a: semi-major axis

b: semi-minor axis

d = a−√a2 − b2: x- oordinate of the rst fo al point

125

dF1 F2a

b

Figure 7.18: Ellipse with extreme point at the origin and the two fo al points F1 and F2.

Equation 7.41 an be rewritten as:

y2 = 2xb2

a− x2

b2

a2(7.42)

If only small values for x are of interest (see above), the se ond term in Eq. 7.42 an be ignored so

that the equation simplies to:

y2 ≈ 2xb2

a(7.43)

If the parameters a and b are hosen in su h a way that b2 = a, the simplied equation for the ellipse

(7.43) orresponds to the simplied equation for the ir le (7.40).

The x- oordinate d of the rst fo al point be omes

d = a−√

a2 − b2 = a−√

a2 − a (7.44)

If - the other way round - the x- oordinate d of the rst fo al point is given, the semi-major axis a is

found as

a =d2

2d− 1(7.45)

Eq. 7.45 reveals for a only positive (valid) solutions, if d > 0.5. For a sour e position with x- oordinatexQ > 0.5, the ree tion at the ir ular ar an thus be approximated as ree tion at an ellipse where

the sour e point orresponds to the rst fo al point and the se ond fo al point is given as:

xF2 = 2x2Q

2xQ − 1− xQ =

xQ2xQ − 1

(7.46)

where:

xF2: x- oordinate of the se ond fo al point

The ree tion at the ellipti ally shaped boundary manifests in su h a way that rays emitted at the rst

fo al point all meet in the se ond fo al point.

Fd

Figure 7.19: Parabola with vertex at the origin and fo al point F .

Parabola The equation that des ribes the parabola in Fig. 7.19 is given by:

y2 = 2px (7.47)

126

where:

p: parameter

d = p2 : x- oordinate of the fo al point

The equation for the parabola (7.47) with p = 1 orresponds dire tly to the equation for the ir le in

the approximation (7.40) for small x. Consequently for a sour e point with xQ = 0.5, the ree tion at

the ar of a ir le an be approximated by the ree tion at a parabola with fo al point at xQ = 0.5.

All rays emitted at the fo al point of a parabola are ree ted ba k in parallel to the x axis.

F1daMitte

Figure 7.20: Hyperbola with vertex at the origin and the rst fo al point F1.

Hyperbola The hyperbola in Fig. 7.20 is des ribed by:

(x+ a)2

a2− y2

b2= 1 (7.48)

where:

a: x-axis parameter

b: y-axis parameter

d = −a+√a2 + b2: x- oordinate of the rst fo al point

The equation for the hyperbola 7.48 an be rewritten as:

y2 = 2xb2

a+ x2

b2

a2(7.49)

Under the assumption of small x values, the se ond term in Eq. 7.49 an be ignored:

y2 ≈ 2xb2

a(7.50)

If the parameters a and b are hosen in su h a way that b2 = a, the approximated equation of the

hyperbola (7.50) orresponds to the approximated equation of the ir le (7.40). The x- oordinate d of

the rst fo al point be omes

d = −a+√

a2 + b2 = −a+√

a2 + a (7.51)

If - the other way round - the x- oordinate d of the rst fo al point is given, the axis parameter a is

found as

a =d2

1− 2d(7.52)

In Eq. 7.52 positive (valid) solutions for a result only if d < 0.5. For a sour e point with xQ < 0.5,the ree tion at an ar of a ir le an be approximated by the ree tion at a hyperbola with the rst

fo al point orresponding to the sour e position and the se ond fo al point at:

xF2 = −2x2Q

1− 2xQ− xQ =

xQ2xQ − 1

(7.53)

where:

xF2: x- oordinate of the se ond fo al point

Sound rays that are emitted at the rst fo al point are ree ted in su h a way that they seem to

originate from the se ond fo al point. A ording to Eq. 7.53 the x- oordinate of the se ond fo al

127

point is always smaller than −xQ whi h implies that the divergen e of the ree tion is weaker than a

ree tion at a plane surfa e.

It should be noted that the equations for the se ond fo al point are identi al for the ellipse and the

hyperbola. Indeed the equation holds even for the parabola in the limiting ondition of xF2 → ∞.

Table 7.5 summarizes the the above ndings for the geometri al ree tion at a ir ular ar .

hyperboli paraboli ellipti al

x

y

r = 1

x

y

r = 1

x

y

r = 1

sour e position: xQ < 0.5 xQ = 0.5 xQ > 0.5ree tion: divergent parallel fo using

se ond fo al point: xF2 =xQ

2xQ−1 ∞ xF2 =xQ

2xQ−1

Table 7.5: Ree tion at a ir ular ar (bold) with radius r = 1 for dierent sour e positions.

Ree tion at ir les: sour e on the ir le k2

If the sour e is lo ated on the ir le k2 (see Fig. 7.16) the ree ting ar of the large ir le orresponds

approximately to a segment of a verti ally orientated ellipse with the rst fo al point at the sour e

position. Thus the ree ted rays all meet at the se ond fo al point. The se ond fo al lies symmetri ally

to the rst fo al point relative to the line a (see Fig. 7.21).

k2

a

Figure 7.21: A sour e point on ir le k2 produ es ree tions that fo us in a point symmetri al to the

sour e position relative to a.

The analyti al investigation follows the onsiderations from above. Again the ree ting ir ular ar an

be approximated for small x values by

y2 ≈ 2x (7.54)

A verti ally orientated ellipse through the origin an be des ribed by Eq. 7.55.

(x− b)2

b2+y2

a2= 1 (7.55)

128

where:

a: major half axis

b: minor half axis

x = b, y = +√a2 − b2: oordinate of the rst fo al point

x = b, y = −√a2 − b2: oordinate of the se ond fo al point

For small values of x, Eq. 7.55 an be approximated by Eq. 7.56.

y2 ≈ 2xa2

b(7.56)

With the ondition a2 = b, Eq. 7.56 orresponds to the equation of the ir le (7.54). This implies that

the ar of the ir le looks like a segment of an ellipse. The fo al points of this ellipse are given by

y2 = x− x2 (7.57)

Eq. 7.57 des ribes a ir le with enter at xZ = 0.5, yZ = 0 and radius = 0.5 (7.58).

(x− 0.5)2+ y2 = 0.25 ⇔ y2 = x− x2 (7.58)

7.3 Room a ousti s of small rooms, wave theoreti al a ous-

ti s

The sound eld in small rooms at low frequen ies is dominated by dis rete resonan es (Eigenfrequen ies)

with low spe tral density. In these situations the methods of statisti al and geometri al a ousti s are

not appli able. The wave nature of sound has to be onsidered expli itly with help of wave theoreti al

room a ousti s.

7.3.1 Wave equation and boundary onditions

The possible sound elds in a room are given by fun tions of sound pressure that fulll the wave equation

as well as the boundary onditions. If one restri ts to sinusoidal time dependen ies, the wave equation

an be repla ed by the Helmholtz equation (1.51) with the omplex, lo ation dependent amplitude

fun tion p:

p+ k2p = 0 (7.59)

where

k = ωc (wave number)

The boundary onditions are dened by the room limiting surfa es. It is assumed that the surfa es are

lo ally rea ting whi h means that they an be spe ied by an impedan e Z given as the ratio of sound

pressure and normal omponent of the sound parti le velo ity on the surfa e.

With Eq. 1.12 it an be written for a point on the surfa e:

∂p

∂n= −ρ∂vn

∂t(7.60)

Inserting the impedan e Z of the surfa e, the sound parti le velo ity in Eq. 7.60 an be eliminated:

1

ρ

∂p

∂n= − 1

Z

∂p

∂t(7.61)

Introdu ing omplex writing for the sinusoidal sound pressure p = pejωt yields

∂p

∂t= pjωejωt (7.62)

Insertion of (7.62) in (7.61) gives

1

ρ

∂p

∂n= − 1

Zpjω (7.63)

129

or

Z∂p

∂n+ jωρp = 0 (7.64)

7.3.2 Solution for re tangular rooms with a ousti ally hard surfa es

Solutions of the wave equations that fulll the boundary onditions an be found analyti ally for a

few spe ial geometries only. One important example is the re tangular room. Rooms with su h a

fundamental shape are often en ountered in real life.

In the following, a re tangular room with dimensions Lx, Ly, Lz a ording to Fig. 7.22 is onsidered.

L

L

L

z

x

y

x

y

z

Figure 7.22: Coordinate system to be used for the dis ussion of the sound eld in a re tangular room

with dimensions Lx, Ly, Lz.

As a simpli ation it is assumed that all surfa es are a ousti ally hard (Z → ∞). With Eq. 7.64 the

boundary onditions read as:

∂p

∂x= 0 for x = 0, x = Lx

∂p

∂y= 0 for y = 0, y = Ly

∂p

∂z= 0 for z = 0, z = Lz (7.65)

All possible sound elds in the re tangular room are given by sound pressure fun tions p(x, y, z) thatfulll the Helmholtz equation (7.59) and the boundary onditions (7.65). In artesian oordinates the

Helmholtz equation reads as

∂2p

∂x2+∂2p

∂y2+∂2p

∂z2+ k2p = 0 (7.66)

As a guess for the solution, the following approa h will be tested:

p(x, y, z) = C cos

(

nxπx

Lx

)

cos

(

nyπy

Ly

)

cos

(

nzπz

Lz

)

(7.67)

where

nx, ny, nz: arbitrary whole number ≥ 0

C: arbitrary onstant

The approa h (7.67) des ribes a eld of standing waves with maxima and minima, depending on

lo ation. As a proof, the approa h is inserted into the Helmholtz equation and in the boundary

130

ondition equations.

Veri ation of the boundary onditions

For that purpose the Eq. (7.67) is dierentiated regarding the oordinates x, y and z. For the x- oordinate this yields:

∂p

∂x= −Cnxπ

Lxsin

(

nxπx

Lx

)

cos

(

nyπy

Ly

)

cos

(

nzπz

Lz

)

(7.68)

→ ∂p

∂x= 0 for nx integer

Veri ation of the Helmholtz equation

Eq. (7.67) is dierentiated two times regarding the oordinates x, y and z:

∂2p

∂x2= −Cn

2xπ

2

L2x

cos

(

nxπx

Lx

)

cos

(

nyπy

Ly

)

cos

(

nzπz

Lz

)

(7.69)

∂2p

∂y2= −C

n2yπ

2

L2y

cos

(

nxπx

Lx

)

cos

(

nyπy

Ly

)

cos

(

nzπz

Lz

)

∂2p

∂z2= −Cn

2zπ

2

L2z

cos

(

nxπx

Lx

)

cos

(

nyπy

Ly

)

cos

(

nzπz

Lz

)

Inserted in (7.66) yields:

C

[

−n2xπ

2

L2x

−n2yπ

2

L2y

− n2zπ

2

L2z

]

cos

(

nxπx

Lx

)

cos

(

nyπy

Ly

)

cos

(

nzπz

Lz

)

+ (7.70)

k2C cos

(

nxπx

Lx

)

cos

(

nyπy

Ly

)

cos

(

nzπz

Lz

)

= 0

The above equation is satised if

k2 =n2xπ

2

L2x

+n2yπ

2

L2y

+n2zπ

2

L2z

(7.71)

In the re tangular room with a ousti ally hare surfa es the Helmholtz equation is only fullled for

dis rete values of the wave number k (so alled Eigenvalues). Ea h positive, whole numbered triple

nx, ny, nz determines with Eq. 7.71 an Eigenvalue. The orresponding fun tion p(x, y, z) is alled

mode.

With

k =2π

λ= 2π

f

c(7.72)

relation (7.71) an be expressed in frequen y f :

f =c

2

√

n2x

L2x

+n2y

L2y

+n2z

L2z

(7.73)

Figure 7.23 shows some examples of sound pressure distributions (modes) in a re tangular room.

All modes have a sound pressure maximum in the orners of the room. Modes with one ni = 0 have a

maximum at the edges while modes with two ni = 0 show a maximum on the orresponding planes.

This is of relevan e for the pla ement of low frequen y absorbers that rea t on sound pressure (plate

or membrane absorbers).

Table 7.6 shows exemplarily the lowest ten Eigenfrequen ies for a small re tangular room with dimen-

sions 4.7×4.1×3.1 m.

131

0 1 2 3 4 5 6

0

1

2

3

4

0 1 2 3 4 5 6

0

1

2

3

4

0 1 2 3 4 5 6

0

1

2

3

4

0 1 2 3 4 5 6

0

1

2

3

4

Figure 7.23: Sound pressure amplitude distribution in a re tangular room for a few modes. The

amplitude is olour oded where red stands for maximum and blue for minimum amplitudes. From left

to right and top to bottom: mode (2,0,0), mode (1,1,0), mode (2,1,0), mode (3,2,0).

Eigenfrequen y [Hz nx ny nz36.2 1 0 0

41.5 0 1 0

54.8 0 0 1

55.0 1 1 0

65.7 1 0 1

68.6 0 1 1

72.3 2 0 0

77.7 1 1 1

82.9 0 2 0

83.4 2 1 0

Table 7.6: The ten lowest Eigenfrequen ies and the orresponding modes for a re tangular room with

dimensions 4.7×4.1×3.1 m.

The frequen y dieren es between the adja ent Eigenfrequen ies are quite large at the low frequen y

end. For in reasing frequen y these dieren es be ome smaller. In [

40

the number Nf of Eigenfre-

quen ies between 0 and the frequen y f [Hz in a re tangular room of volume V [m

3 is estimated

as

Nf ≈ 4π

3V

(

f

c

)3

(7.74)

The density dNf/df (number of Eigenfrequen ies per Hz) at frequen y f is then

dNfdf

≈ 4πV

(

f2

c3

)

(7.75)

If the resonan es overlap, the room modes are no longer isolated and lose their relevan e. For pra ti al

appli ations, a resonan e width of about 1 Hz an be assumed. Evaluation of Eq. 7.75 yields a

orresponding frequen y fS for a density of 1 mode per Hz:

fS ≈ 1800√V

(7.76)

40

Philip M. Morse, Vibration and Sound (1936).

132

fS an be interpreted as lower limiting frequen y, above whi h the investigation of the sound eld with

statisti al or geometri al a ousti s is valid.

7.3.3 Sour e - re eiver transfer fun tion

The above dis ussed modes in a re tangular room represent the sound elds that are allowed by the

room. In a on rete situation the question arises whether a ertain mode an be ex ited. This leads

to the sour e - re eiver transfer fun tion. The mathemati al treatment makes the introdu tion of a

sour e term in the wave equation ne essary

41

. Here only the solution is given. The sound pressure

amplitude p(E,ω) at a re eiver point E, spe ied by the oordinates ex, ey, ez, with volume ex itation

at a sour e point Q given by (qx, qy, qz) and angular frequen y ω is

p(E,ω) ∼ ω∑

n

pn(E)pn(Q)

(ω2 − ω2n)Kn

(7.77)

where

∑

n: sum over all modes

pn(E): omplex sound pressure amplitude for the mode n at point Epn(Q): omplex sound pressure amplitude for the mode n at point Qωn: Eigenfrequen y for the mode nKn: onstant

From Eq. 7.77 follows that a ertain mode n produ es relevant sound pressure at the re eiver E only

if both Q and E are in the vi inity of a pressure maximum. As already mentioned, all modes have a

pressure maximum in the orners of a re tangular room. Thus if a loudspeaker is expe ted to ex ite

all possible modes, it should be pla ed in a orner.

Up to now perfe tly hard surfa es were assumed. In reality all rooms show at least little absorption.

The onsequen es ompared to the above derived results are

• at the resonan e frequen ies only quasi standing waves establish with nite maxima and not

vanishing minima

• the quality of the resonan es in the transfer fun tion is nite (lowering and widening of the peaks).

The bandwidth of a resonan e in the transfer fun tion is a measure for the damping of the orresponding

mode. In a well damped room this bandwidth is typi ally in the order of 5 Hz. The dying away of a

mode an be hara terized by a sort of reverberation time whi h an be estimated a ording to 7.78

42

RT =2.2

B(7.78)

where

RT : reverberation time in se onds

B: bandwidth (at the -3 dB points)

7.3.4 A ousti al design of small rooms

Introdu tion

In small undamped rooms the following a ousti al di ulties are typi al:

• At low frequen ies the transfer fun tion is very uneven due to the low density of resonan es.

Figure 7.24 shows an example.

• At mid and high frequen ies strong ree tions lead to omb lter distortions and errors in the

stereo image. These ee ts are irrelevant if there is no other ontribution stronger than -15 dB

relative to the dire t sound within 20 ms after the dire t sound

43

.

41

H. Kuttru, Room A ousti s, Elsevier, 1991.

42

F. Alton Everest, Master Handbook of A ousti s, M Graw Hill, 2001

43

James A. S. Angus, Controlling Early Ree tions Using Diusion, AES Convention 102nd (1997).

133

• At all frequen ies the reverberation is too large whi h leads to low transparen y of the a ousti al

image.

60 80 100 120 140 160 180 200

frequency [Hz]

−70

−60

−50

−40

−30

−20

−10

0

transfe

r fu

ncti

on [

dB

]

Figure 7.24: Example of a transfer fun tion between a loudspeaker and a mi rophone in an undamped

studio room.

The a ousti al design of a small room has to ensure tat the above mentioned problems are avoided.

There are two fundamental strategies:

• installation of absorbers

• installation of diusers

Absorbers

Low frequen y absorbers for the low frequen y range are typi ally realized as plate or membrane ab-

sorbers. To obtain a broad frequen y band of absorption, dierent modules are ne essary with adjusted

resonan e frequen y. In the mid and high frequen y range porous absorbers an be used.

Diusers

The use of diusers aims at repla ing ree tions by s attering

44

. In the best ase the s attered sound

energy is equally distributed in all dire tions. In small rooms, s attering may help to avoid room

resonan es. In order to reate diuse ree tions a surfa e has to introdu e lo ally inhomogeneous

ree tion onditions. This inhomogeneity an be realized by phase or amplitude variation. An

important ategory of diusers are S hroeder diusers that are based on thorough mathemati al

investigations

45

,

46

. S hroeder diusers are built from a series of narrow hannels of varying depth

(Fig. 7.25). An in ident sound wave that hits the diuser runs down in ea h hannel, is ree ted and

re-emitted at the hannel entran e. The varying hannel depth introdu es a random phase shift that

yields a more or less uniform radiation.

A serious drawba k of this onguration is the relative high absorption asso iated with the ree tion.

This is due to partial sound pressure ompensation of adja ent hannels during radiation. The hannel

on ept an be extended to fra tal stru tures where the primary hannel with and depth is designed

for low frequen ies and the high frequen y s attering is realized by a smaller stru ture at the bottom

of ea h hannel (Fig. 7.26).

A di ulty arises if identi al panels are put in line. Due to the introdu ed periodi ity ertain frequen ies

will be ree ted predominantly in ertain dire tions. This unwanted artefa t an be over ome with the

44

T. J. Cox, P. D'Antonio, A ousti Absorbers and Diusers, Taylor and Fran is, 2009.

45

M. R. S hroeder, Diuse Sound Ree tion by Maximum Length Sequen es, Journal of the A ousti al So iety of

Ameri a, vol. 57, p. 149-150 (1975).

46

M. R. S hroeder, Binaural Dissimilarity and Optimum Ceilings for Con ert Halls: More Lateral Sound Diusion,

Journal of the A ousti al So iety of Ameri a, vol. 65, p.958-963 (1979).

134

usage of dierent panel types. An ex ellent overview of diusers an be found in

47

.

Figure 7.25: Se tion through a S hroeder diuser with hannels of varying depth a ording to a distin t

number sequen e.

Figure 7.26: Example of a fra tal S hroeder diuser.

Depending on the stru ture depth, the frequen y range of a diuser is limited to low frequen ies.

However re ent developments show that it is possible to further lower this limit with help of a tive

strategies

48

.

The determination of the diusivity of a stru ture by measurements an be performed a ording to

the ISO standard 17497-1

49

. The method yields a frequen y independent single gure in form of a

so alled s attering oe ient. The measurement is based on several impulse response measurements

in the reverberation hamber while the stru ture is rotating. By phase sensitive averaging of the re-

sponses, the spe ular ree tion ( oherent ontribution) separates from the diuse ree tion (in oherent

ontribution).

Design of listening rooms

The design of listening rooms an be based on the standard DIN 15996

50

. The standard spe ies

the maximum allowable noise level, the reverberation time and the sound insulation between dier-

ent fa ilities. Listening rooms should be larger than 40 m

3and symmetri al relative to the listening axes.

The maximum allowable noise levels are given by limiting urves in form of third o tave band spe tra.

The noise may not be higher than the limiting values in none of the third o tave bands. An advan ed

listening room should omply with the limit GK10 (Fig. 7.27).

The sound insulation between two dierent listening rooms should be so high that the mutual distur-

ban e lies below the GK10 urve. For this evaluation a listening spe trum a ording to Fig. 7.28 is

assumed.

Depending on the room volume, the reverberation time in the 500 Hz third o tave band should lie

between 0.3 (50 m

3) and 0.5 (1000 m

3) se onds. The reverberation time should be onstant over

frequen y (± 10% in the range from 125 to 2000 Hz).

47

Peter D'Antonio, Trevor Cox, Two De ades of Sound Diusor Design and Development, Part 1: Appli ations and

Design, Journal of the Audio Engineering So iety, vol. 46, no. 11, p.955-976 (1998).

48

Trevor Cox, et al., Maximum length sequen es and Bessel diusers using a tive te hnologies, Journal of Sound and

Vibration, vol. 289, p.807-829 (2006).

49

ISO 17497-1, A ousti s - Sound-s attering properties of surfa es - Measurement of the random-in iden e s attering

oe ient in a reverberation room (2004).

50

DIN 15996, Elektronis he Laufbild- und Tonbearbeitung in Film-, Video- und Rundfunkbetrieben (1996).

135

63

100

160

250

400

630

1000

1600

2500

4000

6300

10000

third-octave band [Hz]

0

10

20

30

40

50

level [d

B]

Figure 7.27: Limiting urve GK10 to spe ify the maximum allowable noise level in third o tave bands.

63

100

160

250

400

630

1000

1600

2500

4000

6300

10000

third-octave band [Hz]

50

60

70

80

90

level [d

B]

Figure 7.28: Assumed sound pressure spe trum in a typi al listening situation.

7.4 Room a ousti al measurements

The traditional measurement quantity in room a ousti s is reverberation time. There are dierent ways

to measure the reverberation time as e.g. with noise that is swit hed-o or by reverse integration of the

squared impulse response. The reverberation time represents a global attribute, in the frame of diuse

eld theory the reverberation time does not depend on sour e and re eiver positions. However in

pra ti al measurements there o ur dieren es for varying positions. Therefore the reverberation time

of a room has to be determined as the average over typi ally two sour e and ve re eiver positions.

Along with the measurements, the air temperature and humidity have to be logged to estimate and

normalize the ee t of air absorption. Further information regarding room a ousti al measurements

an be found in the standard ISO-3382.

In re ent years room impulse response measurements be ome more and more popular. For given

sour e and re eiver positions the impulse response ontains the omplete information of the

room (Fig. 7.29). The main advantage of room impulse responses lies in the possibility to investi-

gate the strength of single ree tions and to evaluate further obje tive riteria su h as larity, EDT, et .

The impulse response and the derived obje tive riteria are very sensitive to the sour e dire tivity. To

get results of general validity an omnidire tional sour e is used. Possible sour es to ex ite a room are

pistol shots or balloon bursts

51

or loudspeakers. However the pra ti al realization of a wide-band,

omnidire tional loudspeaker is di ult. One strategy is to pla e several speaker hassis on a sphere-like

surfa e su h as a dode ahedron (Fig. 7.30).

If an impulse response measurement is performed with a line array of mi rophones, additional information

51

J. Pätynen et al., Investigations on the balloon as an impulse sour e, J. A oust. So . Am., EL27-EL33, vol. 129

(2011).

136

Figure 7.29: Example of a measured impulse response in a multi-purpose hall. The rst peak orresponds

to the dire t sound, followed by weak ree tions at the ground and at nearby obje ts. Later, more

pronoun ed ree tions from the walls and the eiling arrive and nally the reverberation tail an be

observed. From the se tion before the arrival of the dire t sound the unwanted noise and thus the

quality of the measurement an be estimated.

about the sound in iden e dire tion an be obtained

52

. This allows for a more reliable identi ation

of single ree tions.

52

A. J. Berkhout, D. de Vries, J. J. Sonke, Array te hnology for a ousti wave eld analysis in en losures, J. of the

A ousti al So iety of Ameri a, vol. 102, no. 5 (1997), p.2757-2770.

137

Figure 7.30: Dode ahedron loudspeaker with 12 hassis for omnidire tional sound radiation.

138

Chapter 8

Building a ousti s

8.1 Introdu tion

Building a ousti s deals with noise ontrol in buildings. The fundamental aim is the avoidan e or

su ient redu tion of noise from neighbors. Usually there is no onne tion by air between two adja ent

rooms. However air borne or stru ture borne sound in one room nds its way to the other room by

vibration of the stru ture. Finally this vibration is emitted in form of air borne sound in the re eiver

room. The apability of a wall to suppress this transmission is alled sound insulation. Two forms

of ex itation are possible. The rst type of ex itation is air borne sound su h as a talking person or

a loudspeaker. The sound insulation in this ontext is alled airborne sound insulation. The se ond

type is stru ture borne sound whi h means the stru ture is ex ited dire tly by a me hani al for e. The

most important sour e of this type is impa t sound that o urs while walking. In this ase the sound

insulation is alled impa t sound insulation.

8.2 Airborne sound insulation

8.2.1 Sound insulation index R

The airborne sound insulation of a stru ture that separates two rooms (Figure 8.1) is des ribed by the

transmission loss or airborne sound insulation index R a ording to Eq. 8.1.

sender room receiver room

structure under investigation

Figure 8.1: Conguration of a sender and a re eiving room with the separating stru ture to be investi-

gated.

R = 10 log

(

P1

P2

)

[dB (8.1)

where

P1: in ident sound power on the sender side

139

P2: sound power that is radiated on the rear side of the stru ture

The sound insulation index R is independent of the area of the stru ture.

The measurement of R is based on a sound pressure level dieren e L1 − L2 in third o taves between

the sender and re eiving room. However two orre tions have to be applied:

• The level L1 des ribes the sound pressure square in the sender room. Under the assumption

that the sound eld an be thought of as omposed of plane waves arriving from all dire tions,

the in ident sound power P1 an be determined by integration over a half sphere and taking the

osine of the in ident angle into a ount. With S as area of the stru ture the sound power results

as P1 = Sp21/4ρc.

• The sound pressure square p22 in the re eiving room is inverse proportional to the total absorption

A2 in the re eiving room. Consequently the power P2 is given as P2 = A2p22/4ρc. A2 is

determined with the reverberation time T2 and the room volume V2 as A2 = 0.16V2/T2.

Finally for the sound insulation index an be written

R = L1 − L2 + 10 log

(

S

A2

)

[dB (8.2)

Details about the measurement of sound insulation of building elements an be found in the series of

standards ISO 140-3. For easier handling the third o tave spe trum of R is onverted to a single gure

Rw (rated sound insulation index) by appli ation of a referen e spe trum.

8.2.2 Sound insulation of single walls

Sound insulation of homogeneous and dense plates depends on frequen y and the plate parameters:

• thi kness

• density

• modulus of elasti ity

The frequen y dependen y of R follows essentially the urve shown in Figure 8.2. Hereby three regions

A, B and C an be distinguished.

fg d

f d

R

A B

C

Figure 8.2: General frequen y dependen y of the sound insulation index R for a single wall. The

abs issa shows the produ t frequen y × thi kness of the element (= fd). Region A: mass law, region

B: oin iden e, region C: above oin iden e.

Region A:

For low frequen ies the sound insulation follows the mass law that an be written for random in ident

sound waves as

1

R = 20 log

(

πfm′′

ρc

)

− 5 [dB (8.3)

1

Fasold, Sonntag, Winkler, Bau- und Raumakustik, Verlag R. Müller (1987).

140

where

f : frequen ym′′

: area spe i mass

For a given stru ture the sound insulation in reases by 6 dB for a doubling of frequen y. In the same

manner for a given frequen y the sound insulation in reases by 6 dB for a doubling of the mass.

Region B:

The ex itation of the wall by a sound leads to the formation of bending waves. These waves

propagate along the surfa e with a velo ity that depends on the modulus of elasti ity and the

thi kness and density of the stru ture. If the wave length of the airborne sound ex itation on the wall

(proje tion of the wave) oin ides with the wave length of the bending wave, the sound insulation

ollapses. This ondition is alled oin iden e. Exa t oin iden e o urs for a ertain frequen y

and a ertain sound in iden e dire tion. Due to the random distribution of the angle of in iden es

the oin iden e ollapse is not that strong in the diuse eld and smeared over a wider frequen y region.

Region C:

For frequen ies above the oin iden e the sound insulation in reases again with frequen y. The steepness

is around 25 dB/de a de.

8.2.3 Sound insulation of double walls

An improvement of the sound insulation an be a hieved by adding a se ond wall. The spa e between

the walls is usually air. The two walls together with the air spa e in between form a resonan e system

with two masses oupled by a spring. At the resonan e frequen y the sound insulation breaks down and

is lower than in the ase of a orresponding single wall. Above resonan e the sound insulation in reases

strongly with frequen y up to the point where again oin iden e ki ks in.

8.2.4 Standard sound pressure level dieren e

In a given situation the disturban e of neighbors does not depend primarily on the sound insulation

index of the stru tural elements, but rather on the sound pressure level dieren e DnT between the

rooms. This level dieren e is given by the sound insulation index R and the shared area F . As the

sound pressure level in the re eiving room is inuen ed by the total absorption A, an agreement has

to be a hieved to get representative results. This is done by normalizing the results to a reverberation

time in the re eiver room of 0.5 s. For a re eiver room volume V , the standard sound pressure level

dieren e an be written as

DnT = R+ 10 log

(

V

F

)

− 4.9 (8.4)

If the rated sound insulation index Rw is inserted in Eq. 8.4, the orresponding value is alled rated

standard sound pressure level dieren e with the symbol DnT,w.

8.3 Impa t sound insulation

The measurement of the impa t sound insulation is usually based on ex itation by a standardized tapping

ma hine. The ma hine uses hammers of dened mass and form that fall on the oor from dened height.

In the re eiving room the resulting sound pressure level is measured at dierent positions. From the

average sound pressure level Li the standard impa t sound level Ln is determined by normalization for

a total absorption of 10 m

2. With the re eiving room volume V this an be expressed with help of the

reverberation time T in the re eiving room as

Ln = Li − 10 log

(

10T

0.163V

)

(8.5)

The spe tral values Ln an be translated into a single value Ln,w by omparison with a referen e urve.

141

8.4 SIA 181

The Swiss standard SIA 181 represents the state of the art in building te hnology regarding building

a ousti al requirements. The standard denes the ne essary noise prote tion on two levels. The

minimal requirements have to be fullled in any ase. Apart from the minimal requirements elevated

requirements are spe ied that an be agreed by ontra t. In some ases su h as single family houses

that are built together, the elevated requirements are ompulsory.

The SIA 181 denes minimal values of sound pressure level dieren es of the building stru ture for

exterior airborne sound and interior airborne sound. In addition, limiting values are given for impa t

sound. The limiting values dierentiate regarding the intensity of the sour e and the degree of sensitivity

of the inhabitants for a ertain usage of the room.

8.5 Constru tion hints for good building a ousti al onditions

Arrangement of rooms Often building a ousti al problems an be avoided by suitable arrangement

of rooms. It should be avoided that rooms with dierent usage (e.g. a bed room and a kit hen)

are lo ated next to ea h other (horizontally and verti ally).

Doors and windows Doors and windows have typi ally a maximum sound insulation of 35 to 40 dB.

Higher values an only be obtained with spe ial onstru tions. Compared to doors and windows

the sound transmission through the surrounding walls an usually be negle ted.

Leakage The sound insulation between adja ent rooms is drasti ally redu ed if there is leakage in form

of ra ks. Similarly lead-throughs for ables or ventilation du ts are riti al.

Floating oors Usually walls are put dire tly on the on rete oors. To avoid signi ant stru ture

borne sound transmission through the oor, oating oors an be installed. Hereby a layer of low

stiness is put in between the on rete oor and the top over. It is absolutely ru ial that any

onne tion between the oating oor and other parts of the building onstru tion is avoided.

142

Chapter 9

Noise abatement

9.1 Introdu tion - denition of noise

Noise is sound but sound is not ne essarily noise. The assessment of an a ousti al situation regarding

possible annoyan e for a human being depends strongly on the individual. Noise is very subje tive and

as su h an't be measured. Ea h person has his own noise s ale. Furthermore annoyan e depends on

the momentary ondition of the individual (psy hologi al situation, weariness, et .). A short denition

of noise is:

Noise is unwanted sound

Noise has to be assessed, there is no obje tive s ale. For well dened noise sour es su h as road tra

or railways a relation between an obje tive a ousti al measure (exposure) and the annoyan e an be

established. However su h a relation is only valid for an average person, the individual reporting an

deviate signi antly. The out ome of studies about annoyan e follows typi ally a urve as shown in

Fig. 9.1. The sigmoid urve expresses the fa t that even for very low exposure always a ertain portion

of people reports high annoyan e. At the other end, there are very insensitive people that are not

signi antly annoyed even at very high exposure.

20 30 40 50 60 70 80

noise exposure

0

20

40

60

80

100

perc

enta

ge o

f hig

hly

annoyed p

ers

ons

Figure 9.1: Typi al relation between noise exposure and annoyan e, shown as per entage of people that

are highly annoyed.

In the meantime it is widely a epted that ex ess noise may ause health problems. The orresponding

relations are di ult to establish due to the omplexity and number of fa tors that play a role. However

it an be assumed that risk of health impairment due to noise in reases for average sound pressure levels

higher than 65 to 70 dB(A) during the day. At night the sound pressure level at the ear of the sleeping

person should not ex eed 30 dB(A) in order not to ae t sleep quality.

143

9.2 Ee ts of noise

The ee ts of noise an be ategorized as follows:

physiologi al ee ts su h as heada he, ardio-vas ular diseases, in reased blood pressure, extensive

pouring out of stress hormones, sleep disturban es and hearing defe ts in extreme ases

psy hologi al ee ts su h as stress and nervousness, redu tion of produ tivity

so ial ee ts su h as obstru tion of ommuni ation, so ial segregation (those who an aord live in

quieter areas)

In addition to the above mentioned ee ts noise has e onomi al onsequen es as well. The noise burden

is a fa tor that has signi ant inuen e on the pri es of real estates. In many situations measures have

to be taken against noise (su h as e.g. noise barriers). In ase of publi noise sour es (roads, railway

lines, et .) the osts are payed by the publi . Finally noise indu ed health problems ause health osts

and loss of produ tivity.

9.3 General remarks for the assessment of noise

The assessment of noise is usually based on the exposure prin iple. Besides the intensity of the

noise events the number of events in a ertain time interval is taken into a ount. This leads to

the onsideration of average values su h as the Leq (energy equivalent sound pressure level). The

averaging period is often a year.

The sensitivity to noise is highest during nighttime, somewhat lower at the evening period and lowest

during the day. Switzerland has hosen the approa h to dene separate limiting values for day and

night. In Europe and the U.S. the so alled day-evening-night level Lden is used. The Lden maps the

noise exposure to a single number whereby the level for the night period is in reased by 10 dB and the

evening level is in reased by 5 dB. These malus values ree t the in reased sensitivity during night

and evening periods.

Lden = 10 log

(

1

24

[

12 · 100.1(Ld) + 4 · 100.1(Le+5) + 8 · 100.1(Ln+10)]

)

(9.1)

where

Ld: average sound pressure level Leq during daytime (12 hours)

Le: average sound pressure level Leq during the evening period (4 hours)

Ln: average sound pressure level Leq at night (8 hours)

In some ases the day-night level Ldn is used. It is dened analogously to the Lden, however without onsideration of the evening period.

Ldn = 10 log

(

1

24

[

15 · 100.1(Ld) + 9 · 100.1(Ln+10)]

)

(9.2)

where

Ld: average sound pressure level Leq during daytime (7:00 till 22:00)

Ln: average sound pressure level Leq during nighttime (22:00 till 7:00)

The assessment of a noise situation is nally based on a omparison of the exposure at a re eiver lo ation

with a limiting value. This yields a simple yes/no de ision. In addition there exist more sophisti ated

assessment s hemes that evaluate a ontinuous relationship between exposure and annoyan e - an

example is the Zür her Fluglärmindex, ZFI.

9.4 Inuen e of the sour e type

At equal exposure people report dierent annoyan e for dierent noise sour es. Railway noise for

example is signi antly less annoying ompared to road tra noise or noise from air rafts (Fig. 9.2)

1

.

1

M. E. Miedema, H. Vos, Exposure-response relationships for transportation noise, Journal of the A ousti al So iety

of Ameri a, vol. 104, p.3432-3445 (1998).

144

50 55 60 65 70 75 80

Ldn [dB(A)]

0

10

20

30

40

50

60

70

80

perc

enta

ge o

f hig

hly

annoyed p

ers

ons

railwayroadair traffic

Figure 9.2: Exposure - annoyan e relation for dierent noise sour es. The annoyan e is expressed as

per entage of people that are highly annoyed, the exposure is des ribed as Ldn.

The urves in Fig. 9.2 orrespond to the fun tions in Eq. 9.3, where %HA is the per entage of highly

annoyed people.

railwaynoise : %HA = 0.01(Ldn − 42) + 0.0193(Ldn − 42)2

roadtrafficnoise : %HA = 0.03(Ldn − 42) + 0.0353(Ldn − 42)2

aircraftnoise : %HA = 0.53(Ldn − 42) + 0.0285(Ldn − 42)2 (9.3)

There are several reasons for a sour e type dependent annoyan e sensitivity. An important inuen e

fa tor is the personal attitude towards the noise polluter. Furthermore spe tral or temporal dieren es

in the noise signal may play a role. Consequently in pra ti e ea h kind of noise is investigated and

assessed separately.

9.5 Denition of limiting values

As dis ussed above the noise burden is investigated by evaluating a suitable exposure measure and

subsequent omparison with limiting values. The denition of these limiting values is based on exposure

- annoyan e relationships as shown in Fig. 9.1. Usually the annoyan e is reported on a s ale from 0 to

10. The per entage of highly annoyed people is then determined by ounting the answers 8. . .10. Thelimiting value is typi ally set to the exposure that reates between 15 and 25% highly annoyed people.

In other words if the limiting value is rea hed, almost one quarter of the people is highly annoyed.

9.6 Legal basis in Switzerland

9.6.1 Environment prote tion law USG

The environment prote tion law was implemented in 1985. It spe ies the fundamental prin iples for the

prote tion of humans, animals and plants against harmful and annoying impa ts. As a entral instru tion

the prin iple of pre aution was established. It says that potential impa ts should be dete ted in advan e

and limited a ordingly. All emissions should be limited at the sour e a ording to the possibilities given.

The exposure at residents has to be assessed by omparison with impa t thresholds. These limits have

to be xed in su h a way that - a ording to best knowledge - exposures below the limits guarantee that

the population is not sin erely annoyed. The law is further detailed in the Noise Abatement Ordinan e

LSV.

9.6.2 Noise Abatement Ordinan e LSV

The Noise Abatement Ordinan e (LSV) spe ies the exe ution of the environment prote tion law in

the domain of noise. The LSV has been put into for e in 1987 and has experien ed dierent extensions

145

and adaptations sin e. The LSV gives de larations regarding onstru tion, operation and rehabilitation

of fa ilities and regularizes the onstru tion of new buildings with noise sensitive usage.

S heme of limiting values

The LSV spe ies not only impa t thresholds, but planning values and alarm values as well. The

planning values are typi ally 5 dB lower than the impa t thresholds. They ome into play for new

buildings and new fa ilities and implement the prin iple of pre aution. The alarm values on the other

hand (typi ally 5 dB higher than the impa t thresholds) help to identify severe situations with urgent

need for the realization of noise abatement measures. All limiting values are spe ied separately for

day and night periods. Further they are dierentiated a ording to four sensitivity levels. Sensitivity

level I orresponds to spe ial zones for re reation, sensitivity level II qualies zones for living, sensitivity

level III is assigned to zones for living and industry. Sensitivity level IV nally orresponds to zones with

industry only.

Constru tion, operation and sanitation of fa ilities

As a fundamental prin iple the LSV laims that any noise sour e has to redu e its emissions as mu h

as possible at least to a degree that is aordable.

A new or heavily altered installation has to redu e its emissions, so that the planing values in the

neighborhood are respe ted. For private installations relaxations an be granted if the installation is of

general interest or if the eort to fulll the planing values would be disproportional. Publi installations

an get relaxations as well, even if the impa t threshold is violated. However in these ases prote tion

measures have to be taken at the re eivers in form of sound-proof windows.

Existing installations have to respe t the impa t thresholds in the neighborhood. If a private installation

ex eeds these values, the installation has to be improved. Relaxations are possible between the impa t

threshold and the alarm value. Publi installations an get relaxations even above the alarm value if

prote tion measures are taken at the re eivers.

If a private installation is signi antly altered and the impa t thresholds were violated so far, measures

have to be taken to respe t the impa t thresholds.

Constru tion permits

An important aim of the LSV is the prevention that new buildings with noise sensitive usage are

built in areas with high noise burden. Therefor the allowan e for new buildings is oupled to ertain

onditions regarding noise that is already present. The authorities an install new zones for buildings

only if the planing values an be respe ted. Similarly, areas that are already dened as zones for

buildings but are not developed yet have to respe t the planing values. Houses are allowed in zones

for buildings that are already developed if the impa t thresholds are kept. Ex eptions are possible

if the onstru tion is of publi interest, e.g. if a gap in row of houses is losed to reate a quiet ba kyard.

Relevant for the veri ation of the limiting values is the enter of the most exposed open window of a

room with noise sensitive usage su h as living rooms or bed rooms. In the vi inity of line noise sour es

su h as roads or railway lines it may be possible to onstru t new houses even in short distan e if the

orientation of the sensitive rooms is optimized. For windows that an not be opened the noise limits

do not apply.

Assessment of road tra noise

To evaluate the road tra noise burden two rating levels Lr are determined separately for day (6-22)

and night (22-6) as follows:

Lr = Leq +K1 (9.4)

Leq orresponds to the yearly average A-weighted sound pressure level, evaluated for day and night.

The orre tion K1 depends on tra volume. For less than 32 vehi les per hour K1 is -5 dB, for more

than 100 vehi les per hour K1 equals 0 dB.

146

The two rating levels evaluated with Eq. 9.4 are nally ompared with the s heme of limiting values in

Table 9.1.

Sens.level PW-day PW-night IGW-day IGW-night AW-day AW-night

I 50 40 55 45 65 60

II 55 45 60 50 70 65

III 60 50 65 55 70 65

IV 65 55 70 60 75 70

Table 9.1: S heme of limiting values for road tra noise for the sensitivity levels I to IV. PW is the

planning value, IGW is the impa t threshold and AW is the alarm value.

Assessment of railway noise

To evaluate the railway noise burden two rating levels Lr are determined separately for day (6-22) and

night (22-6) as follows:

Lr = Leq +K1 (9.5)

Leq orresponds to the yearly average A-weighted sound pressure level, evaluated for day and night.

The orre tion K1 depends on tra volume. For less than 8 train passages per hour K1 is -15 dB,

for more than 80 passages per hour K1 equals -5 dB. This bonus ree ts the lower sensitivity against

railway noise ompared to road tra noise.

Assessment of industry noise

Noise from industries shows larger variation in hara ter ompared with road tra or railway noise.

Usually industrial noise sour es vary over time signi antly. For that reason the assessment is based on

dierent phases of equal noise hara ter. The rating level is dened for day (7-19) and night (19-7) as

follows

Lr = 10 log(

∑

10(0.1Lri))

(9.6)

where the Lri orrespond to partial rating levels of the individual phases of equal noise hara ter. The

partial rating levels are determined as

Lri = Leqi +K1i +K2i +K3i + 10 log

(

tito

)

(9.7)

where:

Leqi: energy equivalent A-weighted sound pressure level during phase iK1i: sour e type dependent orre tion for phase iK2i: tone orre tion for phase iK3i: impulse orre tion for phase iti: average daily duration of phase i in minutes, where ti =

Ti

BTi: yearly duration of phase i in minutes

B: number of days per year the plant is in servi e

2

to = 720 minutes

The orre tion for the sour e type K1 lies between 5 and 10 dB (10 dB are applied for heating,

ventilation and air ondition installations).

The orre tion for tonal sound is set a ording to the listening impression. If there is no tone (with

a distin t pit h) audible, K2 equals = 0, for weakly audible tones K2 is set to 2, for learly audible

tones K2 is 4 and nally if the signal ontains tones that are strongly audible, K2 is set to 6.

2

In some ases the denition of B is tri ky.

147

The orre tion for impulsive sound is determined subje tively as well. K3 = 0 stands for no audible

impulsiveness, K3 = 2 signies weakly audible impulses, K3 = 4 is for learly audible impulses and

K3 = 6 is for strongly audible impulsiveness.

The day and night rating levels a ording to Eq. 9.6 are ompared to the limiting values for road tra

noise (Table 9.1).

Assessment of noise from shooting ranges

The assessment of noise from 50 m and 300 m shooting ranges is based on a rating level Lr as follows:

Lr = L+K (9.8)

where L orresponds to the average maximum level (A-Fast) of a single shot. The orre tion K for the

number of shots is determined as:

K = 10 log(Dw + 3 ·Ds) + 3 log(M)− 44 (9.9)

where:

Dw: number of half-days with a tivity during the week per year

Ds: number of half-days with a tivity at Sundays per year

M : number of shots red in one year

Finally the rating levels are ompared with the limiting values s heme a ording to Table 9.2. As

shooting ranges operate only during daytime, there are no limiting values for the night period.

Sens.level PW IGW AW

I 50 55 65

II 55 60 75

III 60 65 75

IV 65 70 80

Table 9.2: S heme of limiting values for noise from shooting ranges for the sensitivity levels I to IV.

PW is the planning value, IGW is the impa t threshold and AW is the alarm value.

Assessment of air raft noise

The assessment of air raft noise in the surroundings of the airports Zuri h, Basel and Geneva is based

on separate rating levels for the day period (6-22), the rst hour of the night (22-23), the se ond hour

of the night (23-24) and the last hour of the night (5-6). The level for the day period Lrt is determined

as follows:

Lrt = 10 log(100.1Lrk + 100.1Lrg) (9.10)

Lrk orresponds to the rating level for small aviation. The level is determined as the A-weighted

average sound pressure level for a day with average peak servi e and a orre tion based on the number

of ight operations. Lrg is the A-weighted, yearly average sound pressure level stemming from large

aviation in the period between 6 and 22.

The rating levels for the night hours orrespond dire tly to the A-weighted average sound pressure levels

produ ed by large aviation. The rating levels are nally ompared to the s heme given in Table 9.3.

The impa t thresholds for the se ond and last night hour are identi al to the nighttime values for

road tra noise. However the separate evaluation of hourly values in ase of air raft noise is stri ter

ompared to road tra noise where higher values in one hour are smeared over the whole night period.

9.7 Sounds ape on ept

Standard noise abatement strategies try to lower the A-level at the residents lo ations. However the

potential for attenuation measures in urban environments is usually rather small as lassi al solutions

148

Sens.level PWd IGWd AWd PWn1 IGWn1 AWn1 PWn2l IGWn2l AWn2l

I 53 55 60 43 45 55 43 45 55

II 57 60 65 50 55 65 47 50 60

III 60 65 70 50 55 65 50 55 65

IV 65 70 75 55 60 70 55 60 70

Table 9.3: S heme of limits for air raft noise for the sensitivity levels I to IV. PW is the planning value,

IGW is the impa t threshold and AW is the alarm value. The index d denotes the day period (6-22),

n1 indi ates the rst night hour (22-23), n2l means the se ond and last night hour (23-24, 5-6).

su h as noise barriers are not appli able. Therefore a ousti ians and authorities start to re onsider the

fundamental noise abatement goal. The noise situation of residents an usually be improved by lowering

the A-level but this is not ne essarily the only path to go. Indeed people assess noise annoyan e by

taking into a ount many more fa tors. It seems therefore promising to onsider additional aspe ts

when it omes to future noise abatement poli ies. All relevant aspe ts that ae t noise per eption are

usually summarized and des ribed by the Sounds ape.

A mighty fa tor in this ontext is the fa t that subje tive annoyan e depends on the type of noise

sour e. At identi al A-levels, we are usually more annoyed by man-made sounds ompared to natural

sounds. This oers the possibility to mask unwanted sound by more favored sounds su h as water

sounds

3

.

3

L. Galbrun, T. T. Ali, A ousti al and per eptual assessement of water sounds and their use over road tra noise,

J. A ousti al So iety Ameri a, vol. 133, p. 227-237 (2013)

149

Appendix A

A ousti physi al onstants

A.1 speed of sound in air

temperature [

C speed of sound c [m/s

0 331.3

10 337.3

20 343.2

A.2 density of air at sea level

temperature [

C density of air ρ [kg/m

3

0 1.292

10 1.247

20 1.204

A.3 a ousti impedan e

temperature [

C ρc [Ns/m3

0 428.0

10 420.5

20 413.3

150

Index

A-lter, 52

absorber, 97

absorption, 97

absorption oe ient, 97

a ousti al alibrator, 59

a ousti al holography, 39

adiabati strati ation, 92

airborne sound, 4

airborne sound insulation, 139

airow speaker, 41

angular frequen y, 10

atmospheri absorption, 85

auralization, 121

auto orrelation fun tion, 63

B-lter, 52

bang, 24

basilar membrane, 73

boundary onditions, 9

C-lter, 52

C80, 117

alibrators, 58

ent, 83

enter time, 118

larity, 117

o hlea, 73

o ktail party ee t, 79

oherent sour es, 16

omplex tonal sound, 23

riti al band, 76

riti al distan e, 109

ross orrelation fun tions, 63

ylindri al waves, 15

day-evening-night-level, 144

day-night-Pegel, 144

dB - s ale, 21

degrees of freedom, 55

dira tion, 3

diuse ree tion, 19

diuse sound eld, 107

diusers, 134

dipole radiator, 17

dode ahedron loudspeaker, 136

Doppler ee t, 19

ear, 73

Early De ay Time, 117

Eigenfrequen ies, 129

emission measurements, 48

energy impulse response, 116

environment prote tion law, 145

equally tempered s ale, 83

Equivalent ontinuous sound pressure level, 49

equivalent sour e te hnique, 40

Eyring, 112

FAST time onstant, 50

lters, 51

nite element method, 35

free eld response mi rophone, 58

frequen y, 10

frequen y analysis, 52

frequen y analyzers, 60

frequen y response, 62

Fresnel number, 90

Fresnel zone, 30

Fresnel zones, 123

Gabor pulse, 67

geometri al spreading, 2

ground ee t, 86

head related transfer fun tions, 122

Helmholtz equation, 11

HRTF, 79, 122

Huygens elementary sour es, 30

impedan e, 5

impedan e tube, 101

impulse orre tion, 148

impulse response, 62

in oherent sour es, 16

interferen e, 3

Kir hho - Helmholtz integral, 27

Kundt's tube, 100

L1, 60

L50, 60

Lambert's ree tion hara teristi s, 115

lateral energy fra tion, 118

level re orders, 60

levels, 21

loudness, 74

loudness summation, 76

masking, 76

maximum length sequen es, 64

measurement un ertainty, 54

membrane absorbers, 99

151

membranes, 46

Mi roown, 61

mi roperforated absorbers, 98

mi rophones, 58

mirror sour e, 19

mirror sour es (room a ousti s), 120

MLS, 64

mode, 131

momentary level, 49

noise, 143

Noise Abatement Ordinan e, 145

noise: evaluation of railway noise, 147

noise: evaluation of road tra noise, 146

noise: industrial installations, 147

noise: shooting ranges, 148

o tave lters, 54

organ pipe, 41

otoa ousti emission, 74

period length, 10

phon, 74

pink noise, 23

pistonphone, 58

plane waves, 12

point sour es, 15

pre eden e ee t, 81

pressure response mi rophone, 58

pressure zone onguration, 71

prin iple of re ipro ity, 40

pure tone, 23

quarter wave length resonator, 25

ray tra ing (room a ousti s), 120

Rayleigh integral, 27

ree tion, 2, 19, 97

ree tion oe ient, 97

reverberation time, 68, 111, 116, 136

reverberation time measurement, 68

rods, 45

room impulse response measurement, 136

Sabine, 112

s ale models, 120

s attering, 3, 26

S hroeder reverse integration, 69

S hroeder-diusers, 134

seat dip ee t, 123

SIA 181, 142

siren, 41

SLOW time onstant, 50

sone, 74

soni boom, 20

sound exposure level, 49

sound eld, 4

sound insulation, 139

sound insulation index, 139

sound intensity, 4

sound intensity level, 21

sound intensity meters, 61

sound level meter, 59

sound parti le displa ement, 4

sound parti le velo ity, 4

sound power, 4

sound power level, 21

sound pressure, 4

sound pressure level, 21

sound re orders, 60

sounds ape, 149

sour e dire tivity, 19

spe ular ree tion, 18

spheri al waves, 13

standing wave, 24

strength, 117

strings, 43

superposition of point sour es, 15

superposition prin iple, 15

sweep, 24

system identi ation, 62

temporal masking, 79

thermo-a ousti al ma hines, 46

third-o tave lters, 54

time-bandwidth un ertainty prin iple, 67

time-reversed a ousti s, 40

tone burst, 24

tone orre tion, 147

total absorption, 108

transmission, 97

un ertainty of measurements, 54

virtual pit h, 77

volume velo ity, 5

wave equation, 5, 8

wave equation of a string, 43

wave length, 10

wave number, 10

white noise, 23

152