Impact of Wall Impedance Phase Angle on Indoor Sound Field and Reverberation Parameters Derived from Room Impulse Response
Abstract
Accurate definition of boundary conditions is of crucial importance for room acoustic predictions because the wall impedance phase angle can affect the sound field in rooms and acoustic parameters applied to assess a room reverberation. In this paper, the issue was investigated theoretically using the convolution integral and a modal representation of the room impulse response for complex-valued boundary conditions. Theoretical considerations have been accompanied with numerical simulations carried out for a rectangular room. The case of zero phase angle, which is often assumed in room acoustic simulations, was taken as a reference, and differences in the sound pressure level and decay times were determined in relation to this case. Calculation results have shown that a slight deviation of the phase angle with respect to the phase equal to zero can cause a perceptual difference in the sound pressure level. This effect was found to be due to a change in modal frequencies as a result of an increase or decrease in the phase angle. Simulations have demonstrated that surface distributions of decay times are highly irregular, while a much greater range of the early decay time compared to the reverberation time range indicates that a decay curve is nonlinear. It was also found that a difference between the decay times predicted for the complex impedance and real impedance is especially clearly audible for the largest impedance phase angles because it corresponds approximately to 4 just noticeable differences for the reverberation metrics.Keywords:
room acoustics, complex wall impedance, indoor sound field, room impulse response, reverberation parametersReferences
1. Aretz M., Dietrich P., Vorländer M. (2014), Application of the mirror source method for low frequency sound prediction in rectangular rooms, Acta Acustica united with Acustica, 100(2): 306–319, https://doi.org/10.3813/AAA.918710
2. Collins P.J. (2006), Differential and Integral Equations, Oxford University Press, New York.
3. Cox T., D’Antonio P. (2009), Acoustic Absorbers and Diffusers: Theory, Design, and Application, 2nd ed., Taylor & Francis, New York, https://doi.org/10.4324/9781482266412
4. Damelin S.B., Miller W. Jr (2012), The Mathematics of Signal Processing, Cambridge University Press, New York.
5. Dance S., Van Buuren G. (2013), Effects of damping on the low-frequency acoustics of listening rooms based on an analytical model, Journal of Sound and Vibration, 332(25): 6891–6904, https://doi.org/10.1016/j.jsv.2013.07.011
6. Hak C.C.J.M., Wenmaekers R.H.H.C., van Luxemburg L.C.J. (2012), Measuring room impulse responses: impact of the decay range on derived room acoustic parameters, Acta Acustica united with Acustica, 98(6): 907–915, https://doi.org/10.3813/AAA.918574
7. Hunt F.V., Beranek L.L., Maa D.Y. (1939), Analysis of sound decay in rectangular rooms, The Journal of the Acoustical Society of America, 11(1): 80–94, https://doi.org/10.1121/1.1916010
8. Jeong C.-H. (2012), Absorption and impedance boundary conditions for phased geometrical acoustics methods, The Journal of the Acoustical Society of America, 132(4): 2347–2358, https://doi.org/10.1121/1.4740494
9. Jeong C.-H., Lee D., Santurette S., Ih J.-G. (2014), Influence of impedance phase angle on sound pressures and reverberation times in a rectangular room, The Journal of the Acoustical Society of America, 135(2): 712–723, https://doi.org/10.1121/1.4861839
10. Kuttruff H. (2009), Room Acoustics, 5th ed., Spon Press, New York.
11. Long M. (2014), Architectural Acoustics, 2nd ed., Elsevier Academic Press, New York.
12. Maa D.-Y. (1939), Distribution of eigentones in a rectangular chamber at low frequency range, The Journal of the Acoustical Society of America, 10(3): 235–238, https://doi.org/10.1121/1.1915981
13. Meissner M. (2008), Influence of wall absorption on low-frequency dependence of reverberation time in room of irregular shape, Applied Acoustics, 69(7): 583–590, https://doi.org/10.1016/j.apacoust.2007.02.004
14. Meissner M. (2017), Acoustics of small rectangular rooms: Analytical and numerical determination of reverberation parameters, Applied Acoustics, 120: 111–119, https://doi.org/10.1016/j.apacoust.2017.01.020
15. Meissner M. (2019), Prediction of low-frequency sound field in rooms with complex-valued boundary conditions on walls, Vibrations in Physical Systems, 30(1): 2019127.
16. Meissner M. (2021), Application of modal expansion method for sound prediction in enclosed spaces subjected to boundary excitation, Journal of Sound and Vibration, 500: 116041, https://doi.org/10.1016/j.jsv.2021.116041
17. Meissner M., Wisniewski K. (2020), Investigation of damping effects on low-frequency steady-state acoustical behaviour of coupled spaces, Royal Society Open Science, 7(8): 200514, https://doi.org/10.1098/rsos.200514
18. Meissner M., Zielinski T.G. (2020), Low-frequency prediction of steady-state room response for different configurations of designed absorbing materials on room walls, [in:] Proceedings of the 29th International Conference on Noise and Vibration Engineering (ISMA 2020) and the 8th International Conference on Uncertainty in Structural Dynamics (USD 2020), pp. 463–477.
19. Murphy D.T, Southern A., Savioja L. (2014), Source excitation strategies for obtaining impulse responses in finite difference time domain room acoustics simulation, Applied Acoustics, 82: 6–14, https://doi.org/10.1016/j.apacoust.2014.02.010
20. Navarro J.M., Escolano J. (2015), Simulation of building indoor acoustics using an acoustic diffusion equation model, Journal of Building Performance Simulation, 8(1): 3–14, https://doi.org/10.1080/19401493.2013.850534
21. Nolan M., Davy J.L. (2019), Two definitions of the inner product of modes and their use in calculating non-diffuse reverberant sound fields, The Journal of the Acoustical Society of America, 145(6): 3330–3340, https://doi.org/10.1121/1.5109662
22. Predka E., Branski A. (2020), Analysis of the room acoustics with impedance boundary conditions in the full range of acoustic frequencies, Archives of Acoustics, 45(1): 85–92, https://doi.org/10.24425/aoa.2020.132484
23. Prislan R., Veble G., Svenšek D. (2016), Raytrace modeling of acoustic Green’s function based on the semiclassical (eikonal) approximation, The Journal of the Acoustical Society of America, 140(4): 2695–2702, https://doi.org/10.1121/1.4964295
24. Schroeder M.R. (1965), New method of measuring reverberation time, The Journal of the Acoustical Society of America, 37(3): 409–412, https://doi.org/10.1121/1.1909343
25. Suh J.S., Nelson P.A. (1999), Measurement of transient response of rooms and comparison with geometrical acoustic models, The Journal of the Acoustical Society of America, 105(4): 2304–2317, https://doi.org/10.1121/1.426837
26. Summers J.E. (2012), Accounting for delay of energy transfer between coupled rooms in statistical-acoustics models of reverberant-energy decay, The Journal of the Acoustical Society of America, 132(2): 129–134, https://doi.org/10.1121/1.4734591
27. Szemela K., Rdzanek W. (2022), The influence of an impedance obstacle on the acoustic field inside a rectangular room, Journal of Vibration and Acoustics, 144(2): 021005, https://doi.org/10.1115/1.4051587
28. Winkler-Skalna A., Nowoswiat A. (2021), Use of n-perturbation interval ray tracing method in predicting acoustic field distribution, Applied Mathematical Modelling, 93: 426–442, https://doi.org/10.1016/j.apm.2020.12.028
29. Yasuda Y., Saito K., Sekine H. (2020), Effects of the convergence tolerance of iterative methods used in the boundary element method on the calculation results of sound fields in rooms, Applied Acoustics, 157: 106997, https://doi.org/10.1016/j.apacoust.2019.08.003
30. Yokota T., Sakamoto S., Tachibana H. (2002), Visualization of sound propagation and scattering in rooms, Acoustical Science and Technology, 23(1): 40– 46, https://doi.org/10.1250/ast.23.40
31. Yoshida T., Okuzono T., Sakagami K. (2020), Time domain room acoustic solver with fourth-order explicit FEM using modified time integration, Applied Sciences, 10(11): 3750, https://doi.org/10.3390/app10113750
2. Collins P.J. (2006), Differential and Integral Equations, Oxford University Press, New York.
3. Cox T., D’Antonio P. (2009), Acoustic Absorbers and Diffusers: Theory, Design, and Application, 2nd ed., Taylor & Francis, New York, https://doi.org/10.4324/9781482266412
4. Damelin S.B., Miller W. Jr (2012), The Mathematics of Signal Processing, Cambridge University Press, New York.
5. Dance S., Van Buuren G. (2013), Effects of damping on the low-frequency acoustics of listening rooms based on an analytical model, Journal of Sound and Vibration, 332(25): 6891–6904, https://doi.org/10.1016/j.jsv.2013.07.011
6. Hak C.C.J.M., Wenmaekers R.H.H.C., van Luxemburg L.C.J. (2012), Measuring room impulse responses: impact of the decay range on derived room acoustic parameters, Acta Acustica united with Acustica, 98(6): 907–915, https://doi.org/10.3813/AAA.918574
7. Hunt F.V., Beranek L.L., Maa D.Y. (1939), Analysis of sound decay in rectangular rooms, The Journal of the Acoustical Society of America, 11(1): 80–94, https://doi.org/10.1121/1.1916010
8. Jeong C.-H. (2012), Absorption and impedance boundary conditions for phased geometrical acoustics methods, The Journal of the Acoustical Society of America, 132(4): 2347–2358, https://doi.org/10.1121/1.4740494
9. Jeong C.-H., Lee D., Santurette S., Ih J.-G. (2014), Influence of impedance phase angle on sound pressures and reverberation times in a rectangular room, The Journal of the Acoustical Society of America, 135(2): 712–723, https://doi.org/10.1121/1.4861839
10. Kuttruff H. (2009), Room Acoustics, 5th ed., Spon Press, New York.
11. Long M. (2014), Architectural Acoustics, 2nd ed., Elsevier Academic Press, New York.
12. Maa D.-Y. (1939), Distribution of eigentones in a rectangular chamber at low frequency range, The Journal of the Acoustical Society of America, 10(3): 235–238, https://doi.org/10.1121/1.1915981
13. Meissner M. (2008), Influence of wall absorption on low-frequency dependence of reverberation time in room of irregular shape, Applied Acoustics, 69(7): 583–590, https://doi.org/10.1016/j.apacoust.2007.02.004
14. Meissner M. (2017), Acoustics of small rectangular rooms: Analytical and numerical determination of reverberation parameters, Applied Acoustics, 120: 111–119, https://doi.org/10.1016/j.apacoust.2017.01.020
15. Meissner M. (2019), Prediction of low-frequency sound field in rooms with complex-valued boundary conditions on walls, Vibrations in Physical Systems, 30(1): 2019127.
16. Meissner M. (2021), Application of modal expansion method for sound prediction in enclosed spaces subjected to boundary excitation, Journal of Sound and Vibration, 500: 116041, https://doi.org/10.1016/j.jsv.2021.116041
17. Meissner M., Wisniewski K. (2020), Investigation of damping effects on low-frequency steady-state acoustical behaviour of coupled spaces, Royal Society Open Science, 7(8): 200514, https://doi.org/10.1098/rsos.200514
18. Meissner M., Zielinski T.G. (2020), Low-frequency prediction of steady-state room response for different configurations of designed absorbing materials on room walls, [in:] Proceedings of the 29th International Conference on Noise and Vibration Engineering (ISMA 2020) and the 8th International Conference on Uncertainty in Structural Dynamics (USD 2020), pp. 463–477.
19. Murphy D.T, Southern A., Savioja L. (2014), Source excitation strategies for obtaining impulse responses in finite difference time domain room acoustics simulation, Applied Acoustics, 82: 6–14, https://doi.org/10.1016/j.apacoust.2014.02.010
20. Navarro J.M., Escolano J. (2015), Simulation of building indoor acoustics using an acoustic diffusion equation model, Journal of Building Performance Simulation, 8(1): 3–14, https://doi.org/10.1080/19401493.2013.850534
21. Nolan M., Davy J.L. (2019), Two definitions of the inner product of modes and their use in calculating non-diffuse reverberant sound fields, The Journal of the Acoustical Society of America, 145(6): 3330–3340, https://doi.org/10.1121/1.5109662
22. Predka E., Branski A. (2020), Analysis of the room acoustics with impedance boundary conditions in the full range of acoustic frequencies, Archives of Acoustics, 45(1): 85–92, https://doi.org/10.24425/aoa.2020.132484
23. Prislan R., Veble G., Svenšek D. (2016), Raytrace modeling of acoustic Green’s function based on the semiclassical (eikonal) approximation, The Journal of the Acoustical Society of America, 140(4): 2695–2702, https://doi.org/10.1121/1.4964295
24. Schroeder M.R. (1965), New method of measuring reverberation time, The Journal of the Acoustical Society of America, 37(3): 409–412, https://doi.org/10.1121/1.1909343
25. Suh J.S., Nelson P.A. (1999), Measurement of transient response of rooms and comparison with geometrical acoustic models, The Journal of the Acoustical Society of America, 105(4): 2304–2317, https://doi.org/10.1121/1.426837
26. Summers J.E. (2012), Accounting for delay of energy transfer between coupled rooms in statistical-acoustics models of reverberant-energy decay, The Journal of the Acoustical Society of America, 132(2): 129–134, https://doi.org/10.1121/1.4734591
27. Szemela K., Rdzanek W. (2022), The influence of an impedance obstacle on the acoustic field inside a rectangular room, Journal of Vibration and Acoustics, 144(2): 021005, https://doi.org/10.1115/1.4051587
28. Winkler-Skalna A., Nowoswiat A. (2021), Use of n-perturbation interval ray tracing method in predicting acoustic field distribution, Applied Mathematical Modelling, 93: 426–442, https://doi.org/10.1016/j.apm.2020.12.028
29. Yasuda Y., Saito K., Sekine H. (2020), Effects of the convergence tolerance of iterative methods used in the boundary element method on the calculation results of sound fields in rooms, Applied Acoustics, 157: 106997, https://doi.org/10.1016/j.apacoust.2019.08.003
30. Yokota T., Sakamoto S., Tachibana H. (2002), Visualization of sound propagation and scattering in rooms, Acoustical Science and Technology, 23(1): 40– 46, https://doi.org/10.1250/ast.23.40
31. Yoshida T., Okuzono T., Sakagami K. (2020), Time domain room acoustic solver with fourth-order explicit FEM using modified time integration, Applied Sciences, 10(11): 3750, https://doi.org/10.3390/app10113750
