Properties of Sound Making Waves. Sound Waves ■Sound is created by vibrations.
When Sound Waves meet Solid Surfaces
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When Sound WavesmeetSolid Surfaces
Applications of wave phenomena in room acoustics
By Yum Ji CHANMSc (COME) candidateTU Munich
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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
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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
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Ranges of frequency under interest
Piano
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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
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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
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1.3 Diffraction & Interference Edge interference due to finite plates Reflection on flat surface: Deviation
from ray-like behaviour
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1.4 Fresnel zone Imagine each beam shown below have
pathlengths differered by λ/2 What happens if…
Black + Green? Black + Green + Red?
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1.5 Conclusion drawn from experiment Theory for reflectors in sound is more
complicated than those for light Sizing is important for reflectors
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2.0 Elements controlling sound in a room Reflectors Diffusers Absorbers
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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
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2.2 Size of Reflectors Never too small
Diffraction Absorption
No need to be too big Imagine a mirror for light!
Example worksheet
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2.3 Diffusers Scattering waves With varied geometries
Type 1
Type 2
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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
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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)
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2.6 Absorption at other frequency ranges (B) Low frequency absorbers
Plate absorbers, make use of bending waves
Composite board resonators (VPR in German)
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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)
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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
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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
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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
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3 Parameters in room acoustics Reverberation time Clarity / ITDG (Initial time delay gap) Binaural parameter
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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)
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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
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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
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3.4 Binaural parameter Feel of
spaciousness The difference of
sound heard by left and right ears
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3.5 Applications: Reverberant room
Finding the optimum positions of resonators in the test room
(Source: My Master’s thesis)
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3.5.1 Application: Reverberant room Mesh size 0.2 m ~ 30000 degrees of freedom Largest error of eigenvalue ~ 2%
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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)
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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
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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
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3.6 Are these all? Amount of parameters are increasing Models are still necessary to be built
for “acoustic delicate” rooms Concert halls
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3.7 A failed example New York Philharmonic hall
Models were not built Size of reflectors
(Source: Spektrum der Wissenschaft)
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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
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4.2 1D Example 100 m long tube, unity cross section Mesh size 1 m, 2 m and 4 m
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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
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4.3 Numerical error Possible, but not significant if precision of storage
type is enough
1 01000 1
1 0.0011000 1
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5 Conclusion Is acoustics a science or an art?`
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