When Sound Waves
meet
Solid Surfaces
Applications of wave phenomena in room acoustics
By Yum Ji CHAN
MSc (COME) candidate
TU 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)
 p 

SPL  20  log 
p 
 ref 
 Unit: decibel (dB), pref = 2 × 10-5 Pa
 Acoustic power
 More parameters are necessary in
noise measurements (out of the
scope)
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
dv
p1  p2  M
dt
 Mass is the determining factor at a wide
frequency range
 Transmitted energy (i.e. Absorption in
rooms) is higher
p2  M  2u  k
 At low frequencies
 When the plate is not heavy enough
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 springmass 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
(Owner of the resonator: Müller-BBM GmbH)
(Source: My Master’s thesis)
2.7 Asymmetric FE matrices
 FE matrices are usually symmetric
 Maxwell-Betti theorem
 Coupling conditions make matrices
asymmetric
K SS





K SS
K SF
K FF
  w  M SS
 w  
  i   
  pi  


K FF   p  
M SS
M FS
M FF
   F 
 w
 w
i   0 

    
  pi   0 

 
M FF   p   w
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)
Characteristic
eigenfrequency
of the resonator
C
U
0
50
100
150
200
250
Eigenfrequency (Hz)
(Source: My Master’s thesis)
300
2.8 Why is it like that?
 Consider Rayleigh coefficient
T
w
Kw Compression
2
 R T
w Mw Vibration
 Compare increase of PE to increase of
KE
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)
Direct sound
First reflections (early sound)
1
2
3
4
Reverberation
Time
Time
(Courtesy of Prof. G. Müller)
3.2 Reverberation time
 The most important parameter in general applications
 Definition: SPL drop of 60 dB
 pt T60 
  60
20  log 
p
 t 0 
 Formula drawn by Sabine
T60 
0.161  V
S
 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)
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
Direct sound
First reflections (early sound)
12
3
4
Reverberation
Time
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%
 Reverberation time
 The effect of amount
of resonators
Response (dB ref 1e5)
3.5.2 Impulse response
function
60
60
50
50
40
40
30
30
20
20
10
10
0
0
0
0.5
1
1.5
2
2.5
3
3.5
 The effect of internal
damping inside
resonators
Response (dB ref 1e5)
Time (s)
60
60
50
50
40
40
30
30
20
20
10
10
0
0
0
0.5
1
1.5
2
2.5
3
3.5
Time (s)
(Source: My Master’s thesis)
3.5.3 Getting impulse response
functions
 Convolution
 “Effect comes after excitation”
 Mathematical expression

yt    x ht   d
0
 Expression in Fourier (frequency) domain
Y(f) = X(f) H(f)
 X(f) = 1 for impulse
 H(f) = Impulse response function
in time domain
3.5.3 Getting impulse response
functions
 Frequency domain
1.E+08
1.E+07
Response
1.E+06
1.E+05
1.E+04
1.E+03
1.E+02
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
 Time domain
Response (dB ref 1e5)
Frequency (Hz)
60
60
50
50
40
40
30
30
20
20
10
10
0
0
0
0.5
1
1.5
Time (s)
2
2.5
3
3.5
150
160
170
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
2
1

p
2
 p 2 2
c t
c
Po 
o
 Discretization using linear shape functions
 Variable describing acoustic strength
 Corresponding force variables
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
7.0%
6.0%
Error
5.0%
4.0%
3.0%
2.0%
1.0%
0.0%
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
Eigenmode order
100 elements
50 elements
25 elements
4.3 Numerical error
 Possible, but not significant if precision of storage
type is enough
0
 1


1000
1


0.001
 1


1000
1


5 Conclusion
 Is acoustics a science or an art?`