The WAPE model proposed here is an implementation of the work of [Ostashev et al 2020]. The WAPE is developed using a Padé(1,1) series expansion. A Crank-Nicholson algorithm is used to reduce the equation into a matrix system. Then, the 2D acoustic pressure field is calculated using a second-order centered finite difference scheme on velocity potential.
This code is intended to be shared with the research community.
This WAPE model is designed to run with Matlab, without any additonal toolbox.
To run a calculation you need to open "initialisation_WAPE.m", set parameters values, and run the script. It is indicated which parameter can be modified and which can not.
The input parameters are:
%% Source parameters
freq = 250; % source frequency (Hz)
hS = 35; % source height (m)
%% Medium properties
T = 10; % atmospheric temperature at the surface (°C)
Tlog = 0.2; % temperature coefficient for the vertical gradient
shear_exp = 0.15; % wind shear exponents (scalar) for power law wind profil
v_ref = 5; % wind speed (m/s) measured at z_ref height, for power-law wind profile
z_ref = 80; % reference height for wind speed v_ref
theta = 0; % propagation angle with respect to the source (0° : downwind, 180° : upwind)
f_turb_ind = true; % logical 'true' or 'false' to account for turbulence or not
gamT = 0; % turbulence strength
%% 2D Spatial domain (x,z)
dim_x = 3000; % horizontal dimension x of the domain (m)
dim_z = 300; % vertical dimension z of the domain (m)
haut_a = 0.8; % where the absorbing layer starts at the top of the domain, according to z axis (0.8 ==> start at 80% of z axis)
coeff_a = 10; % damping rate
discrx = 10; % discretization x = lambda/discrx
discrz = 10; % discretization z = lambda/discrz
z_r = 1.5; % receiver height for plotting (m)
%% Ground properties
hv = 0; % vegetation height (m), it affects the shape of atmospheric profils
z0 = 0.13*hv + 0.00001; % atmospheric roughness length (m), can't be null
d = 0.66*hv; % displacement height of flux profiles (m), it's directly linked to vegetation height
lc = 0 ; % correlation length (m) (ground roughness parameter), if 0 : no ground rugositiy, [0.05-1]
sigmah = 0; % standard deviation of roughness height (m) (ground roughness parameter), [0.01-0.05]
sigma = 10000000; % airflow resistivity of the ground (kN.s.m-4)
h = 0; % thickness of ground surface layer (m), if h = 0 no layerWhen the simulation has completed there will be 'DeltaL.mat' and 'normalized_SPL.mat' files which correspond respectively to attenuation to free field (dB), and sound pressure level field (dB) which is normalized by the maximum amplitude. You can post-process these signals to your liking. Here is an exemple of output:
Instabilities can occur with PE simulations if:
- the spatial steps are too large,
- the absorbent layer at the top of the domain is not thick enough.
Calculation time can be really long and take a lot of memory if:
- the spatial steps are too small,
- the domain is too big,
- the frequency is too high.
This code is released under the EUPL license.
The development of this code took part into the french PIBE project (contract ANR-18-CE04-0011).
If this WAPE script contributes to an academic publication, please cite it as:
@misc{kayser_wide_2023,
title = {Wide-angle parabolic equation model},
url = {https://github.com/bkayser13/WAPE/},
author = {Kayser, Bill},
year = {2023},
note = {https://github.com/bkayser13/WAPE/},
}
This code largely implements algorithms that have already been published. A non-exhaustive list is proposed below.
Tappert, F. D. (1977). The parabolic approximation method. Wave propagation and underwater acoustics, 224-287.
Gilbert, K. E., & White, M. J. (1989). Application of the parabolic equation to sound propagation in a refracting atmosphere. The Journal of the Acoustical Society of America, 85(2), 630-637.
West, M., Gilbert, K., & Sack, R. A. (1992). A tutorial on the parabolic equation (PE) model used for long range sound propagation in the atmosphere. Applied Acoustics, 37(1), 31-49.
Collins, M. D. (1993). A split‐step Padé solution for the parabolic equation method. The Journal of the Acoustical Society of America, 93(4), 1736-1742.
Ostashev, V. E., Juvé, D., & Blanc-Benon, P. (1997). Derivation of a wide-angle parabolic equation for sound waves in inhomogeneous moving media. Acta Acustica united with Acustica, 83(3), 455-460.
Dallois, L., Blanc-Benon, P., & Juvé, D. (2001). A wide-angle parabolic equation for acoustic waves in inhomogeneous moving media: Applications to atmospheric sound propagation. Journal of Computational Acoustics, 9(02), 477-494.
Lihoreau, B., Gauvreau, B., Bérengier, M., Blanc-Benon, P., & Calmet, I. (2006). Outdoor sound propagation modeling in realistic environments: Application of coupled parabolic and atmospheric models. The Journal of the Acoustical Society of America, 120(1), 110-119.
Cheinet, S. (2012). A numerical approach to sound levels in near-surface refractive shadows. The Journal of the Acoustical Society of America, 131(3), 1946-1958.
Kayser, B., Gauvreau, B., & Ecotière, D. (2019). Sensitivity analysis of a parabolic equation model to ground impedance and surface roughness for wind turbine noise. The Journal of the Acoustical Society of America, 146(5), 3222-3231.
Ostashev, V. E., Muhlestein, M. B., & Wilson, D. K. (2019). Extra-wide-angle parabolic equations in motionless and moving media. The Journal of the Acoustical Society of America, 145(2), 1031-1047.
Ostashev, V. E., Wilson, D. K., & Muhlestein, M. B. (2020). Wave and extra-wide-angle parabolic equations for sound propagation in a moving atmosphere. The Journal of the Acoustical Society of America, 147(6), 3969-3984.