When Sound WavesmeetSolid Surfaces

Applications of wave phenomena in room acoustics

By Yum Ji CHANMSc (COME) candidateTU Munich

0 Introduction Phemonena of sound waves Equipments on surfaces to control

sound intensity Applications in room acoustics Numerical aspects of finite element

method in acoustics Conclusion

1.0 Nature of sound Sounds are mechanical waves Sound waves have much longer wavelength

than light Speed of sound in air c ≈ 340m/s Wavelength for sound λ

c = f · λ When f = 500 Hz, λ = 68 cm

Typical wavelength of visible light= 4-7 × 10-7 m

Conclusion Rules for waves more important than rules for

rays

Ranges of frequency under interest

Piano

1.1 Measurement of Sound intensity Acoustic pressure in terms of sound

pressure level (SPL)

Unit: decibel (dB), pref = 2 × 10-5 Pa Acoustic power More parameters are necessary in

noise measurements (out of the scope)

refppSPL log20

1.2 Huygen’s principle From wikipedia:

It recognizes that each point of an advancing wave front is in fact the center of a fresh disturbance and the source of a new train of waves; and that the advancing wave as a whole may be regarded as the sum of all the secondary waves arising from points in the medium already traversed.

Diffraction & Interference apply

1.3 Diffraction & Interference Edge interference due to finite plates Reflection on flat surface: Deviation

from ray-like behaviour

1.4 Fresnel zone Imagine each beam shown below have

pathlengths differered by λ/2 What happens if…

Black + Green? Black + Green + Red?

1.5 Conclusion drawn from experiment Theory for reflectors in sound is more

complicated than those for light Sizing is important for reflectors

2.0 Elements controlling sound in a room Reflectors Diffusers Absorbers

2.1 Weight of Reflectors Newton’s second law of motion:

Difference in acoustic pressure = acceleration

Mass is the determining factor at a wide frequency range

Transmitted energy (i.e. Absorption in rooms) is higher At low frequencies When the plate is not heavy enough

dtdvMpp 21

22p M u k

2.2 Size of Reflectors Never too small

Diffraction Absorption

No need to be too big Imagine a mirror for light!

Example worksheet

2.3 Diffusers Scattering waves With varied geometries

Type 1

Type 2

2.4 Absorbers Apparent solution: Fabrics and porous

materials Reality: it is effective only at HF range Needed in rooms where sound should be

damped heavily (e.g. lecture rooms) Because clothes are present

Other absorbers make use of principles in STRUCTURAL DYNAMICS

2.5 Absorption at other frequency ranges (A) Hemholtz

resonator-based structures Analogus to spring-

mass system Example worksheet The response

around resonant frequency depends on damping

Draw energy out of the room

(Source: http://physics.kenyon.edu/EarlyApparatus/index.html)

2.6 Absorption at other frequency ranges (B) Low frequency absorbers

Plate absorbers, make use of bending waves

Composite board resonators (VPR in German)

2.7 Comparison between a composite board resonator and a plate VPR Resonator assembly Modelled as a fluid-solid coupled

assembly with FE Asymmetric FE matrices

(Source: My Master’s thesis)

(Owner of the resonator: Müller-BBM GmbH)

2.7 Asymmetric FE matrices FE matrices are usually symmetric

Maxwell-Betti theorem Coupling conditions make matrices

asymmetric

w

F

ppww

ppww

i

i

FF

FFFS

SS

SS

i

i

FF

FF

SFSS

SS

00

MMM

MM

KKKK

K

2.7 Comparison between a composite board resonator and a plate Bending waves without air backing (Uncoupled, U) Compressing air volume with air backing (Coupled, C)

(Source: My Master’s thesis)

0 50 100 150 200 250 300

U

C

Eigenfrequency (Hz)

Characteristiceigenfrequencyof the resonator

2.8 Why is it like that? Consider Rayleigh coefficient

Compare increase of PE to increase of KE

2T

TR w Kww Mw Vibration

Compression

3 Parameters in room acoustics Reverberation time Clarity / ITDG (Initial time delay gap) Binaural parameter

3.1 Impulse response function of a room The sound profile after an impulse (e.g.

shooting a gun or electric spark in tests)

Time

Direct sound

First reflections (early sound)

Reverberation

1 2

34E

n er gy

Time

(Courtesy of Prof. G. Müller)

3.2 Reverberation time The most important parameter in general applications Definition: SPL drop of 60 dB

Formula drawn by Sabine

Depends on volume of the room and “the equivalent absorptive area” of the room

Samples to listen: Rooms with extremely long RT: Reverberant room

(Courtesy of Müller-BBM)

SVT

161.0

60

60log200

60

t

Tt

pp

3.3 Clarity / ITDG Clarity: Portion of

early sound (within 80 ms after direct sound) to reverberant sound

ITDG: Gap between direct sound and first reflection, should be as small as possible

Time

Direct sound

First reflections (early sound)

Reverberation

1 234

Energy

Time

3.4 Binaural parameter Feel of

spaciousness The difference of

sound heard by left and right ears

3.5 Applications: Reverberant room

Finding the optimum positions of resonators in the test room

(Source: My Master’s thesis)

3.5.1 Application: Reverberant room Mesh size 0.2 m ~ 30000 degrees of freedom Largest error of eigenvalue ~ 2%

3.5.2 Impulse response function

Reverberation time The effect of amount

of resonators

The effect of internal damping inside resonators

0

10

20

30

40

50

60

0 0.5 1 1.5 2 2.5 3 3.5

Time (s)

Resp

onse

(dB

ref 1

e5)

0

10

20

30

40

50

60

0

10

20

30

40

50

60

0 0.5 1 1.5 2 2.5 3 3.5

Time (s)

Resp

onse

(dB

ref 1

e5)

0

10

20

30

40

50

60

(Source: My Master’s thesis)

3.5.3 Getting impulse response functions Convolution

“Effect comes after excitation” Mathematical expression

Expression in Fourier (frequency) domainY(f) = X(f) H(f)

X(f) = 1 for impulse

H(f) = Impulse response functionin time domain

0 dthxty

3.5.3 Getting impulse response functions Frequency domain

Time domain

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170

Frequency (Hz)

Res

pons

e

0

10

20

30

40

50

60

0 0.5 1 1.5 2 2.5 3 3.5

Time (s)

Resp

onse

(dB

ref 1

e5)

0

10

20

30

40

50

60

3.6 Are these all? Amount of parameters are increasing Models are still necessary to be built

for “acoustic delicate” rooms Concert halls

3.7 A failed example New York Philharmonic hall

Models were not built Size of reflectors

(Source: Spektrum der Wissenschaft)

4.1 Acoustic problems with the finite element (FE) method Wave equation

Discretization using linear shape functions

Variable describing acoustic strength Corresponding force variables

22

2 2

1 ppc t

o

o

Pc

4.2 1D Example 100 m long tube, unity cross section Mesh size 1 m, 2 m and 4 m

4.2 1D Example Discretization error in diagram

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

6.0%

7.0%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Eigenmode order

Erro

r

100 elements 50 elements 25 elements

4.3 Numerical error Possible, but not significant if precision of storage

type is enough

1 01000 1

1 0.0011000 1

5 Conclusion Is acoustics a science or an art?`