When Sound Waves meet Solid Surfaces

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When Sound WavesmeetSolid Surfaces

• Applications of wave phenomena in room acoustics
• By Yum Ji CHAN
• MSc (COME) candidate
• TU 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 ?
• c f ?
• When f 500 Hz, ? 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)

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1.2 Huygens 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 ?/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

• Newtons 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

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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
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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

(Owner of the resonator Müller-BBM GmbH)
(Source My Masters thesis)
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2.7 Asymmetric FE matrices

• FE matrices are usually symmetric
• Maxwell-Betti theorem
• Coupling conditions make matrices asymmetric

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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 Masters thesis)
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2.8 Why is it like that?

• Consider Rayleigh coefficient
• Compare increase of PE to increase of KE

Compression
Vibration
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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)

(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)
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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

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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 Masters 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

(Source My Masters thesis)
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3.5.3 Getting impulse response functions

• Convolution
• Effect comes after excitation
• Mathematical expression
• Expression in Fourier (frequency) domain
• Y(f) X(f) H(f)
• X(f) 1 for impulse
• H(f) Impulse response functionin time domain

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3.5.3 Getting impulse response functions

• Frequency domain
• Time domain

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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

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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

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4.3 Numerical error

• Possible, but not significant if precision of
  storage type is enough

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5 Conclusion

• Is acoustics a science or an art?

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