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Applications of Sound Wave Phenomena in Room Acoustics

This paper explores the phenomenon of sound waves and their interactions with solid surfaces, emphasizing their applications in room acoustics. It discusses key concepts such as sound intensity measurement, Huygens’ principle, and the roles of reflectors, diffusers, and absorbers in controlling sound in various environments. The paper also examines numerical approaches, particularly finite element methods in acoustics, to provide a deeper understanding of sound behavior in spaces. Concluding insights reveal the complexities of sound reflection and absorption necessary for effective room acoustics design.

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Applications of Sound Wave Phenomena in Room Acoustics

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  1. When Sound WavesmeetSolid Surfaces Applications of wave phenomena in room acoustics By Yum Ji CHAN MSc (COME) candidate TU Munich

  2. 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

  3. 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

  4. Ranges of frequency under interest Piano

  5. 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)

  6. 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

  7. 1.3 Diffraction & Interference • Edge interference due to finite plates • Reflection on flat surface: Deviation from ray-like behaviour

  8. 1.4 Fresnel zone • Imagine each beam shown below have pathlengths differered by λ/2 • What happens if… • Black + Green? • Black + Green + Red?

  9. 1.5 Conclusion drawn from experiment • Theory for reflectors in sound is more complicated than those for light • Sizing is important for reflectors

  10. 2.0 Elements controlling sound in a room • Reflectors • Diffusers • Absorbers

  11. 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

  12. 2.2 Size of Reflectors • Never too small • Diffraction • Absorption • No need to be too big • Imagine a mirror for light! • Example worksheet

  13. Type 2 2.3 Diffusers • Scattering waves • With varied geometries Type 1

  14. 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

  15. 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)

  16. 2.6 Absorption at other frequency ranges (B) • Low frequency absorbers • Plate absorbers, make use of bending waves • Composite board resonators (VPR in German)

  17. 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)

  18. 2.7 Asymmetric FE matrices • FE matrices are usually symmetric • Maxwell-Betti theorem • Coupling conditions make matrices asymmetric

  19. Characteristiceigenfrequencyof the resonator 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)

  20. 2.8 Why is it like that? • Consider Rayleigh coefficient • Compare increase of PE to increase of KE Compression Vibration

  21. 3 Parameters in room acoustics • Reverberation time • Clarity / ITDG (Initial time delay gap) • Binaural parameter

  22. Direct sound First reflections (early sound) Energy 1 2 3 4 Reverberation Time Time 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)

  23. 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)

  24. Direct sound First reflections (early sound) Energy 1 2 3 4 Reverberation Time Time 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

  25. 3.4 Binaural parameter • Feel of spaciousness • The difference of sound heard by left and right ears

  26. 3.5 Applications: Reverberant room • Finding the optimum positions of resonators in the test room (Source: My Master’s thesis)

  27. 3.5.1 Application: Reverberant room • Mesh size 0.2 m • ~ 30000 degrees of freedom • Largest error of eigenvalue ~ 2%

  28. 3.5.2 Impulse response function • Reverberation time • The effect of amount of resonators • The effect of internal damping inside resonators (Source: My Master’s thesis)

  29. 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

  30. 3.5.3 Getting impulse response functions • Frequency domain • Time domain

  31. 3.6 Are these all? • Amount of parameters are increasing • Models are still necessary to be built for “acoustic delicate” rooms • Concert halls

  32. 3.7 A failed example • New York Philharmonic hall • Models were not built • Size of reflectors (Source: Spektrum der Wissenschaft)

  33. 4.1 Acoustic problems with the finite element (FE) method • Wave equation • Discretization using linear shape functions • Variable describing acoustic strength • Corresponding force variables

  34. 4.2 1D Example • 100 m long tube, unity cross section • Mesh size 1 m, 2 m and 4 m

  35. 4.2 1D Example • Discretization error in diagram

  36. 4.3 Numerical error • Possible, but not significant if precision of storage type is enough

  37. 5 Conclusion • Is acoustics a science or an art?`

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