1. Simulation of the acoustics of coupled rooms by numerical resolution of a diffusion equation V. Valeau a, J. Picaut b, A. Sakout a, A. Billon a a LEPTAB, University of La Rochelle, France b LCPC, Nantes, France 18 th International Congress on Acoustics Kyoto - 04/05/2004

2. n The diffuse field assumption in closed spaces assumes that sound energy is uniform in the field. Model presentation (1) n This is wrong especially for complex closed spaces or long rooms Diffusion equation for acoustic energy density w ( room mean free path, c sound speed) Diffusion coefficient with n Recent works [Picaut et al, Acustica 83, 1997] proposed an extension of the concept of diffuse sound field: n This concept allows non-uniform energy density

3. n Scope of this work: – solving numerically the diffusion equation with a FEM solver (Femlab); – application for coupled rooms, for evaluating: n stationary responses; n impulse responses; – comparison with statistical theory-based results. n It has been applied successfully analytically for 1-D long rooms or streets [Picaut et al., JASA 1999] wall (  ) n Sound absorption at walls is taken into account by an exchange coefficient [Picaut et al., Appl. Acoust. 99]: Model presentation (2)

4. Room boundary (Fourier condition) Source Room volume V: Modeling room acoustics with a diffusion equation

5. Source room (1) Coupled room (2) sound source Coupling aperture E1E1 E2E2 mean energy densities Power balance for the two rooms : coupling factor Statistical theory model for coupled rooms: stationary state

6. Stationary response for a 10*10m room Simulation of coupled rooms acoustics: stationary case (1) Shape definition Sound source Meshing

7. Stationary response for a 10*10m room Simulation of coupled rooms acoustics: stationary case (2) Problem definition Dirichlet boundary cond. w=Q Fourier boundary cond. (absorption) dB FEM calculation

8. Stationary response for a 10*10m room Example : Sound distribution at height 1 m S 12 = 6 m 2 - (uniform) - k=0.16 Y (m) dB 46 50 54 -505 X=0 X=10

9. Sound decay model for coupled rooms with statistical theory n Power balance for the two rooms : n Damping constants : mean coupling factor

10. Stationary response for a 10*10m room Simulation of coupled rooms acoustics: sound decay Initial condition w(t 0 ) = w 0 Fourier boundary cond. (absorption)

11. Stationary response for a 10*10m room Sound decay for two identically damped rooms Volumes V 1 =150 m 3, V 2 =100 m 3 Uniform absorption Mean coupling factor 0123 65 70 75 80 85 Time (s) dB room 1 o o o room 2 Statistical theory Diffusion model

12. Stationary response for a 10*10m room Sound decay for a damped room coupled with a reverberant room Volumes V 1 =125 m 3, V 2 =125 m 3 Absorption Mean coupling factor

13. Potential application: acoustics of networks of rooms dB

14. Conclusive remarks n Numerical solving of the diffusion equation with application to diffuse sound field calculation of coupled rooms, with a low computational cost. n Good agreement with statistical theory results n Advantages : n provides fine description of spatial n variation of sound levels and decays. n low computational cost n provides results for arbitrary shapes n Future work: n direct field contribution; n application to networks of rooms. n validation by comparisons with n measurements