When Sound Waves meet Solid surfaces
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Transcript When Sound Waves meet Solid surfaces
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
pi 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
20 log
pt 0
60
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
yt x ht 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?`