HAL Id: hal-03235294
https://hal.archives-ouvertes.fr/hal-03235294
Submitted on 26 May 2021
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
FDTD Modelling of Sound Propagation in Air Including
Viscothermal and Relaxation Effects
Brian Hamilton, Stefan Bilbao
To cite this version:
FDTD MODELLING OF SOUND PROPAGATION IN AIR INCLUDING
VISCOTHERMAL AND RELAXATION EFFECTS
Brian Hamilton
University of Edinburgh
Acoustics & Audio Group
brian.hamilton@ed.ac.uk
Stefan Bilbao
University of Edinburgh
Acoustics & Audio Group
sbilbao@ed.ac.uk
ABSTRACT
Wave-based modelling has the potential to be a complete solution to virtual acoustic simulation as it is valid across the entire audible frequency range – as opposed to tradi-tional approaches based on geometric principle which are, in general, high-frequency approximations. While wave-based methods have traditionally been applied to low fre-quencies, their use in mid and high frequencies is be-coming increasingly possible thanks to advances in high-performance and parallel computing. In such frequency ranges the effects of sound absorption in air become sig-nificant and cannot be neglected from simulations intended for auralization. In this paper, a system of equations relat-ing to linear acoustics includrelat-ing viscothermal and relax-ation effects in air is devised for the purposes of large-scale simulation of homogeneous urban/outdoor and in-door (room) acoustics, and for potential use in virtual real-ity. A discrete model is constructed for the free-space prob-lem using finite-difference time-domain (FDTD) methods, which is validated through numerical simulations and com-parisons against analytical models.
1. INTRODUCTION
Wave-based modelling of room acoustics is an active topic of research and has applications in virtual acoustics and virtual reality [1]. The finite-difference time-domain (FDTD) method is a popular simulation technique for wave-based modelling of room acoustics [2, 3] because of its simplicity and ability to take advantage of parallel pro-cessing on graphics propro-cessing unit (GPU) devices [4, 5].
Sound absorption in air including viscothermal and re-laxation effects has previously been modelled with FDTD in the general non-linear case [6, 7]. In the context of room acoustics, the lossless second-order wave equation is a popular starting model for efficiency reasons [8], to which air absorption effects have been simulated using dig-ital filters in a post-processing step [5, 9]. Modal/domain-decomposition wave-based methods are also available, which can take into account arbitrary dispersion and sorption profiles (and thus classical and relaxation air ab-sorption effects) through direct access to modal frequency and loss coefficients [10]. Towards a more direct simu-lation of the physical mechanisms underlying air absorp-tion, the viscothermal wave equation has been simulated
with FDTD and implemented on high-performance paral-lel computing devices (GPUs) [4, 11]. The work presented here can be seen as an extension of that previous work through the addition of relaxation effects, with the aim of remaining suitable for large-scale simulations targeting, potentially, the full bandwidth of audible frequencies.
2. A MODEL FOR LINEAR ACOUSTICS IN AIR A simplified model for linear acoustics in air with vis-cothermal and relaxation effects in the absence of sources and boundaries is:
∂t2p − c 2 ∇2p − cη∇2∂tp + 2 π X ν (ανλ)m∂t2pν = 0 (1a) ∂tpν+ 1 τν (pν− p) = 0 (1b) Here, p = p(t, x) is pressure as a function of time t and position x ∈ R3, ∂
t is a partial derivative w.r.t. t, and
∇2 is the 3-D Laplacian. Constants pertaining to air
in-clude: c, the speed of sound; η, a characteristic length scale related to viscothermal effects; (ανλ)m, is maximum
ab-sorption per wavelength due to relaxation effects from a ν-type molecule (oxygen and nitrogen) and τν its
associ-ated relaxation time. Finally, pν = pν(t, x) is an
inter-nal variable with units of pressure associated to a ν-type molecule. The system of equations (1) is reduced from a more complete set of linear acoustics equations for air (with viscothermal and relaxation effects) [12], and arrived at through reasonable assumptions appropriate for room acoustics (including those of zero-mean and irrotational flow). A full derivation is left out for brevity.
Without loss of generality, assume a plane-wave solu-tion of the form p = ei(kx−ωt)with angular frequency ω and wavenumber k. This leads to dispersion relation:
k2= ω 2 c2 + η ciωk 2+ 2 π ω2 c2 X ν (ανλ)m 1 − iωτν (2)
which has the approxiate solution [12]: k = ω c + i ω2η 2c2 + 1 π ω c X ν (ανλ)m 1 − iωτν (3)
The absorption coefficient is given by ={k}, with units Nepers/m, and the phase velocity is given by ω/<{k}.
3. FDTD SCHEME Consider a grid function approximation pn
l u p(nT , lX)
with integer n ≥ 0, l = (lx, ly, lz) ∈ Z3, time-step T and
Cartesian grid spacing X. Define the shift operators: et±pnl = p
n±1
l , ew±pnl = p n
l±ˆew (4)
where ˆewis a standard unit vector and w ∈ {x, y, z}. Then
define temporal and spatial difference operators:
δt±= ±(et±− 1)/T , δw±= ±(ew±− 1)/X (5)
along with
δtt= δt+δt−, δww= δw+δw− (6)
One possible explicit finite difference scheme for (1) is: δttpnl − c 2δ ∆pnl − cηδ∆δt−pnl + 2 π X ν (ανλ)mδttpnν,l= 0 (7a) δt+pnν,l+ 1 τν µt+pnν,l− µt+pnl = 0 (7b) where δ∆ = δxx+ δyy + δzz, pnν,l u pν(nT, lX), and
µt+= (et++ 1)/2. Although left out for brevity, a
stabil-ity analysis over the full space yields the stabilstabil-ity condition T ≤ (pX2/3 + η2− η)/c, which is independent of
relax-ation effects.
It is worth mentioning that while the above scheme is novel (to the authors knowledge), it is similar in con-struction to previous work in the modelling of acoustic tubes [13] and linear string vibration [14], particularly in that the auxiliary equations (1b) are approximated using the trapezoid rule. Such an implicit method may be ap-plied to part of the system without disturbing the resulting explicit character of the scheme as a whole.
4. VALIDATION
A numerical experiment is performed to validate the FDTD scheme (7) against the analytic model (1) in terms of dis-tance attenuation of pressure waves, through dispersion relation (3). Without loss of generality, and in order to simulate a wide range of frequencies, the test simula-tion is limited to an x-direcsimula-tional planar wave on a one-dimensional periodic domain. Physical air constants are chosen from [15] for 20◦C and 20% relative humidity (c ≈ 340 m/s). The spatial step for the scheme is set to X ≈ 0.85 mm and the time-step to T ≈ 1.44 µs. Initial conditions are set from a Gaussian-shaped pressure dis-turbance with variance (0.82 mm)2travelling rightward at speed c. The FDTD scheme is run for a 0.15 s duration on a 34 m domain. The model and the measured FDTD distance attenuations are shown in Figure 1.
As expected by consistency of the FDTD scheme with the model, a good agreement between the numerical results and the model can be observed in low frequencies; and as is typical, numerical dispersion (i.e., FDTD approximation errors) causes discrepancies in high frequencies.
ck/2π (in Hz) 102 103 104 105 d B / m 10−5 10−4 10−3 10−2 10−1 100 FDTD Model
Figure 1. Distance attenuation in dB/m as a function of frequency in Hz, from model (3) and from FDTD simula-tion with scheme (7), for the case of 20◦C and 20% relative humidity.
5. CONCLUSIONS
A linear model for sound propagation in air, including relaxation and viscothermal effects, was provided. This model can be seen as an extension of viscothermal wave equation models previously investigated for large-scale room acoustics simulations. A finite-difference time-domain scheme was composed for this model and validated in terms of distance attenuation of a planar wave for one-dimensional propagation. Future work could include an in-depth analysis of numerical dispersion and stability, and the setting of room-acoustic boundary conditions.
6. ACKNOWLEDGEMENTS
This research was funded in part by ERC grant numbers ERC-StG-2011-279068-NESS, ERC-PoC-2016-WRAM and EPSRC grant number EP/R005001/1.
7. REFERENCES
[1] L. Savioja and N. Xiang, “Introduction to the spe-cial issue on room acoustic modeling and auralization,” Journal of the Acoustical Society of America, vol. 145, no. 4, pp. 2597–2600, 2019.
[2] D. Botteldooren, “Acoustical finite-difference time-domain simulation in a quasi-Cartesian grid,” Journal of the Acoustical Society of America, vol. 95, pp. 2313– 2319, 1994.
[3] K. Kowalczyk and M. van Walstijn, “Room acous-tics simulation using 3-D compact explicit FDTD schemes,” vol. 19, no. 1, pp. 34–46, 2011.
[4] C. J. Webb and S. Bilbao, “Computing room acous-tics with CUDA - 3D FDTD schemes with boundary losses and viscosity,” in Proceedings of the IEEE In-ternational Conference on Acoustics, Speech, and
nal Processing (ICASSP), (Prague, Czech Republic), pp. 317–320, 2011.
[5] J. Saarelma, J. Califa, and R. Mehra, “Challenges of distributed real-time finite-difference time-domain room acoustic simulation for auralization,” in AES In-ternational Conference on Spatial Reproduction - Aes-thetics and Science, July 2018.
[6] M. S. Wochner, A. A. Atchley, and V. W. Sparrow, “Numerical simulation of finite amplitude wave prop-agation in air using a realistic atmospheric absorption model,” The Journal of the Acoustical Society of Amer-ica, vol. 118, no. 5, pp. 2891–2898, 2005.
[7] N. Jim´enez, F. Camarena, J. Redondo, V. S´anchez-Morcillo, Y. Hou, and E. E. Konofagou, “Time-domain simulation of ultrasound propagation in a tissue-like medium based on the resolution of the nonlinear acous-tic constitutive relations,” Acta Acusacous-tica united with Acustica, vol. 102, no. 5, pp. 876–892, 2016.
[8] L. Savioja, T. J. Rinne, and T. Takala, “Simulation of room acoustics with a 3-D finite difference mesh,” (Danish Institute of Electroacoustic Music, Denmark), pp. 463–466, 1994.
[9] J. Huopaniemi, L. Savioja, and M. Karjalainen, “Mod-eling of reflections and air absorption in acoustical spaces a digital filter design approach,” pp. 4–pp, IEEE, 1997.
[10] J. Botts and L. Savioja, “Extension of a spectral time-stepping domain decomposition method for disper-sive and dissipative wave propagation,” Journal of the Acoustical Society of America, vol. 137, no. 4, pp. EL267–EL273, 2015.
[11] B. Hamilton, S. Bilbao, and C. J. Webb, “Improved fi-nite difference schemes for a 3-D viscothermal wave equation on a GPU,” in Proceedings of Forum Acus-ticum, (Krakow, Poland), Sept. 2014.
[12] A. D. Pierce, Acoustics, ch. Effects of Viscosity and Other Dissipative Processes. Springer, Cham, 2019. [13] S. Bilbao and R. Harrison, “Passive time-domain
nu-merical models of viscothermal wave propagation in acoustic tubes of variable cross section,” Journal of the Acoustical Society of America, vol. 140, no. 1, pp. 728– 740, 2016.
[14] C. Desvages and S. Bilbao, “Optimised passive discrete-time models of frequency-dependent loss in linear strings,” in Proceedings of the International Congress on Sound and Vibration (ICSV), (Montr´eal, Canada), July 2019.
[15] International Organization for Standardization, “Acoustics – Attenuation of sound during propagation outdoors. Part 1: Calculation of the absorption of sound by the atmosphere,” Standard ISO 9613–1, Geneva, Switzerland, 1993.
