Post on 07-Feb-2016
description
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?`