1. ASA San Diego 1 Reverberated sound field modelling in coupled rooms using a diffusion equation Alexis Billon a, Vincent Valeau a, Anas Sakout a, Judicaël Picaut b a LEPTAB, University of La Rochelle, France b LCPC, Nantes, France 148 th Meeting of Acoustical Society of America San Diego, 17th november 2004

2. ASA San Diego 2 Model Presentation (1) The diffuse field assumption in closed spaces assumes that sound energy is uniform in the field. 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 Recent works [Picaut et al, Acustica 83,1997] proposed an extension of the concept of diffuse sound field: This concept allows non-uniform energy density Diffusion Model Modelling Statistical theory Positions of the source and the coupling aperture Global sound absorption Source room absorption Synthesis Conclusion

3. ASA San Diego 3 Model Presentation (2) Scope of this work: application for coupled rooms, for evaluating: – stationary responses; – impulse responses; comparison with statistical theory-based results. It has been applied successfully analytically for 1-D long rooms or streets [Picaut et al., JASA 1999] wall (  ) Sound absorption at walls is taken into account by an exchange coefficient [Picaut et al., Appl. Acoust. 99]: Diffusion Model Modelling Statistical theory Positions of the source and the coupling aperture Global sound absorption Source room absorption Synthesis Conclusion

4. ASA San Diego 4 Modelling room acoustics with a diffusion equation (1) Room boundary (Fourier type condition) Source Source roomReceiving room DRhRDRhR DShSDShS Diffusion Model Modelling Statistical theory Positions of the source and the coupling aperture Global sound absorption Source room absorption Synthesis Conclusion

5. ASA San Diego 5 Modelling room acoustics with a diffusion equation (2) Simulated geometry (dimensions in cm) Simulations characteristics: - Unstructured mesh with about 3000 nodes; - stationnary responseSound intensity Level Computing time: less than 1 minute; - impulse responseSound decay Computing time: less than 8 minutes. Diffusion Model Modelling Statistical theory Positions of the source and the coupling aperture Global sound absorption Source room absorption Synthesis Conclusion

6. ASA San Diego 6 Statistical theory model of coupled rooms Source room ( S ) Receiving room ( R ) sound source Coupling aperture EsEs ERER mean energy densities Power balance for the two rooms coupling factor 0< k R <1 Energy decay [Cremer &Müller, 1978] Diffusion Model Modelling Statistical theory Positions of the source and the coupling aperture Global sound absorption Source room absorption Synthesis Conclusion

7. ASA San Diego 7 Effect of source and coupling aperture positions Diffusion Model Modelling Statistical theory Positions of the source and the coupling aperture Global sound absorption Source room absorption Synthesis Conclusion Aperture position Source position The diffusion model is able to depict the non-uniform energy repartition within rooms.

8. ASA San Diego 8 Effect of varying the acoustic absorption coefficient of both rooms (uniform coefficients) Reverberation Time Sound pressure difference Black: diffusion model; red: statistical theory The models show good agreement when the acoustic absorption coefficient of both rooms is varied. Diffusion Model Modelling Statistical theory Positions of the source and the coupling aperture Global sound absorption Source room absorption Synthesis Conclusion

9. ASA San Diego 9 Effect of varying the acoustic absorption coefficient of the source room (receiving room absorption = 0.1) Reverberation Time Sound decay Black: diffusion model; red: statistical theory The diffusion model is able to reproduce the double-decay phenomenon occurring when the receiving room is more reverberant than the source room. Diffusion Model Modelling Statistical theory Positions of the source and the coupling aperture Global sound absorption Source room absorption Synthesis Conclusion

10. ASA San Diego 10 Synthesis of the results STUDIED PARAMETER SOUND PRESSURE DIFFERENCEREVERBERATION TIME TrendHighest discrepancyTrendHighest discrepancy Coupling aperture position similar2.0 dBsimilar3.9% Source positionsimilar2.0 dBsimilar3.9% Global absorption coefficient similar2.8 dBsimilar12.5% Source room absorption coefficient similar2.1 dBsimilar30% Receiving room absorption coefficient similar2.9 dBsimilar21.7% Area of the coupling aperture similar6.1 dBsimilar3.6% Diffusion Model Modelling Statistical theory Positions of the source and the coupling aperture Global sound absorption Source room absorption Synthesis Conclusion

11. ASA San Diego 11 Conclusion The diffusion model shows good agreement with the statistical theory in evaluating: - the sound intensity difference between the rooms; - the reverberation time. Work to come: Comparison with experimental data in coupled rooms, and network of rooms. Acknowledgements: The authors would like to thank the ADEME to support this work. Diffusion Model Modelling Statistical theory Positions of the source and the coupling aperture Global sound absorption Source room absorption Synthesis Conclusion