Acoustics: Wave, Ray, and Statistical Methods

1. Sound field descriptors Eng.Ivaylo Hristev

2. Contents 1. Wave acoustics. Room resonances. 2. Ray acoustics.Raytracing. 3.Statistical acoustics. Reverberation time.

3. Schroeder frequency Division of the audible range. Source:Davis&Davis, 1987

4. 1.Wave acoustics. Wave equation. 3 dimensional wave equation General solution Private solution Private solution with eigenfunctions in 3D

5. For rectangular rooms with hard walls, the private solutions are described as modes ( standing waves ) and are described with the following formula: • nx , nyandnzare whole number parameters; • lx , lyandlzare the room dimensions. • Each three parameters represent room resonance.

6. Standing waves – room modes

7. Standing waves – room modes Sound pressure distribution in a room for axial mode (0,2,0)

8. Standing waves – room modes Sound pressure distribution in a room for tangential mode (1,1,0) Wave propagation Pressure distribution

9. Standing waves – room modes Sound pressure distribution in a room with tangential mode(2,1,0)

10. Standing waves – room modes mode(1,0,0) mode(1,1,0) mode(?,?,0)

11. Standing waves – room modes mode(1,0,0) mode(1,1,0) mode(3,2,0)

12. Room modes in 3 dimensions Axial modes (nx,0,0) (0,ny,0) (0,0,nz) Oblique modes (nx, ny, nz) Tangential modes (nx, ny,0) (nx, 0,nz) (0, ny, nz)

13. Real life measurements

14. Real life measurements Different mode types have different energy and decay in a different way. Tangential modes have ½ of the energy of the axial modes; oblique modes have¼ of the energy of the axial modes.

15. Room mode excitation

16. Non rectangular room modes

17. 2.Ray acoustics. This method makes use of an approximation of the sound wave with a ray. This is analogous to the light rays. By doing so it is straightforward to trace the path of the wave using Snell’s law.

18. Raytracing algorithms Source: http://www.cs.princeton.edu/~funk/acoustics.html

19. 3. Statistical acoustics This method uses only statistical properties of sound field in a room. It is assumed that the sound field is diffuse; this is not true in every room, but results from the theory are within reasonable practical limits.

20. Reverberation

21. Reverberation time Reverberation time can be calculated by the well known Sabine formula: Т60is the time it takes for the sound energy to decay 1000000 times, or sound pressure level decrease of 60 dВ. After this time the sound is considered inaudible.

22. The total sound absorption А is the sum of the product of the sound absorption coefficient by the area of each surface in the room. It is called metric Sabines. А is the sum of the sound absorption of any additional objects, such as people or furniture.

23. The modified formula is known as Eyring formula, gives better results in rooms with large absorption:

24. Sound absorption of air When calculating Rt for big rooms, the air absorption must be accounted for: Heremis the air absorption coefficient.

25. Reverberation time measurements Measuring Rt requires the generation of sound field within the room of interest. The stimulus is then interrupted and the sound decay process is recorded. Usually the measurement stimulus is a special type of noise, and the analysis is done in 1/3 octave bands. The recorded sound should be at least 20 dB above noise. This is rarely possible, that’s why usually the measurements are done by following the decay rate for 10dB (T10), 20 dB (T20) or30dB (Т30) and the result is extrapolated to 60 dB.

26. Reverberation time measurements 1000Hz 500Hz 2000Hz

27. EDT, T30

28. Reverberation time measurements Averaged reverberation time as function of frequency

29. Critical radius Close to the sound source, the sound pressure drops by 6 dBfor distance double. In rooms, the zone for which this holds is within the so called critical radius, at which the direct sound and reverberant field equalize. Beyond this distance – the reverberant field dominates: Here В is the room constant calculated by: