Phase-preserving narrow- and wide-angle parabolic equations for sound propagation in moving mediaa).

Parabolic equations are among the most popular numerical techniques in many fields of physics. This article considers extra-wide-angle parabolic equations, wide-angle parabolic equations, and narrow-angle parabolic equations (EWAPEs, WAPEs, and NAPEs, respectively) for sound propagation in moving inhomogeneous media with arbitrarily large variations in the sound speed and Mach number of the (subsonic) wind speed. Within their ranges of applicability, these parabolic equations exactly describe the phase of the sound waves and are, thus, termed the phase-preserving EWAPE, WAPE, and NAPE. Although variations in the sound speed and Mach number are often relatively small, omitting the second-order terms pertinent to these quantities can result in large cumulative phase errors for long propagation ranges. Therefore, the phase-preserving EWAPE, WAPE, and NAPE can be preferable in applications. Numerical implementation of the latter two equations can be performed with minimal modifications to existing codes and is computationally efficient. Numerical results demonstrate that the phase-preserving WAPE and NAPE provide more accurate results than the WAPE and NAPE based on the effective sound speed approximation.

Wind turbine sound propagation: Comparison of a linearized Euler equations model with parabolic equation methods.

Noise generated by wind turbines is significantly impacted by its propagation in the atmosphere. Hence, for annoyance issues, an accurate prediction of sound propagation is critical to determine noise levels around wind turbines. This study presents a method to predict wind turbine sound propagation based on linearized Euler equations. We compare this approach to the parabolic equation method, which is widely used since it captures the influence of atmospheric refraction, ground reflection, and sound scattering at a low computational cost. Using the linearized Euler equations is more computationally demanding but can reproduce more physical effects as fewer assumptions are made. An additional benefit of the linearized Euler equations is that they provide a time-domain solution. To compare both approaches, we simulate sound propagation in two distinct scenarios. In the first scenario, a wind turbine is situated on flat terrain; in the second, a turbine is situated on a hilltop. The results show that both methods provide similar noise predictions in the two scenarios. We find that while some differences in the propagation results are observed in the second case, the final predictions for a broadband extended source are similar between the two methods.

Sonic boom propagation over real topography.

The effect of elevation variation on sonic boom reflection is investigated using real terrain data. To this end, the full two-dimensional Euler equations are solved using finite-difference time-domain techniques. Numerical simulations are performed for two ground profiles of more than 10 km long, extracted from topographical data of hilly regions, and for two boom waves, a classical N-wave, and a low-boom wave. For both ground profiles, topography affects the reflected boom significantly. Wavefront folding due to terrain depression is notably highlighted. For the ground profile with mild slopes, the time signals of the acoustic pressure at the ground are, however, only slightly modified compared to the flat reference case, and the associated noise levels differ by less than 1 dB. With steep slopes, the contribution due to wavefront folding has a large amplitude at the ground. This results in an amplification of the noise levels: a 3 dB increase occurs at 1% of the positions along the ground surface, and a maximum of 5-6 dB is reached near the terrain depressions. These conclusions are valid for the N-wave and low-boom wave.

Sonic boom reflection over urban areas.

Sonic boom propagation over urban areas is studied using numerical simulations based on the Euler equations. Two boom waves are examined: a classical N-wave and a low-boom wave. Ten urban geometries, generated from the local climate zone classification [Stewart and Oke (2012), Bull. Am. Meteorol. Soc. 93(12), 1879-1900], are considered representative of urban forms. They are sorted into two classes, according to the aspect ratio of urban canyons. For compact geometries with a large aspect ratio, the noise levels and the peak pressure, especially for the N-wave, are highly variable between canyons. For open geometries with a small aspect ratio, these parameters present the same evolution in each urban canyon, corresponding to that obtained for isolated buildings. A statistical analysis of the noise levels in urban canyons is then performed. For both boom waves, the median of the perceived noise levels mostly differs by less than 1 dB from the value obtained for flat ground. The range of variation is greater for open geometries than for compact ones. Finally, low-frequency oscillations, associated with resonant modes of the canyons, are present for both compact and open geometries. Their amplitude, frequency and decay rate vary greatly from one canyon to another.

Sonic boom reflection over an isolated building and multiple buildings.

Sonic boom reflection is investigated over an isolated building and multiple buildings using numerical simulations. For that, the two-dimensional Euler equations are solved using high-order finite-difference techniques. Three urban geometries are considered for two boom waves, a classical N-wave and a low-boom wave. First, the variations of the pressure waveforms and the corresponding perceived noise are analyzed along an isolated building. The influence of the building is limited to an illuminated region at its front and a shadow region at its rear, whose size depends on the building's height and the Mach number. Two buildings are then considered. In addition to arrivals related to reflection on the building facades or to diffraction at the building corners, low-frequency oscillations, associated with resonances, are noticed in the street canyon. Their amplitude depends on the street width and on the incident boom frequency contents. Despite their significance, these low-frequency oscillations have little impact on the perceived noise. Finally, a periodic distribution of identical buildings is examined. The duration of the waveforms is notably increased due to multiple diffraction and canyon resonances. Variations in perceived noise at ground level are moderate for large streets, but become noticeable as the street width reduces.

Influence of meteorological conditions and topography on the active space of mountain birds assessed by a wave-based sound propagation model.

The active space is a central bioacoustic concept to understand communication networks and animal behavior. Propagation of biological acoustic signals has often been studied in homogeneous environments using an idealized circular active space representation, but few studies have assessed the variations of the active space due to environment heterogeneities and transmitter position. To study these variations for mountain birds like the rock ptarmigan, we developed a sound propagation model based on the parabolic equation method that accounts for the topography, the ground effects, and the meteorological conditions. The comparison of numerical simulations with measurements performed during an experimental campaign in the French Alps confirms the capacity of the model to accurately predict sound levels. We then use this model to show how mountain conditions affect surface and shape of active spaces, with topography being the most significant factor. Our data reveal that singing during display flights is a good strategy to adopt for a transmitter to expand its active space in such an environment. Overall, our study brings new perspectives to investigate the spatiotemporal dynamics of communication networks.

Characterization of topographic effects on sonic boom reflection by resolution of the Euler equations.

The influence of topography on sonic boom propagation is investigated. The full two-dimensional Euler equations in curvilinear coordinates are solved using high-order finite-difference time-domain techniques. Simple ground profiles, corresponding to a terrain depression, a hill, and a sinusoidal terrain, are examined for two sonic boom waves: a classical N-wave and a low-boom. Ground reflection of the sonic boom is affected by elevation variations: a concave ground profile induces compression, which tends to increase the peak pressure in particular, while the opposite is true for convex elevation variations, which lead to expansion and a reduction in peak pressure. The reflected boom is then strongly altered. Furthermore, a sufficiently concave topography can cause focal zones, which generate extra contributions at ground level in the form of U-waves in addition to the reflected wave. This mechanism has the largest effect on waveforms at ground level. The variations of standard metrics are of a few dBs compared to a flat ground for both sonic boom waves, and they are notably greater for the terrain depression than for the hill. Finally, in the case of a sinusoidal terrain, the pressure waveforms are composed of multiple arrivals due to successive focal zones.

Effect of surface roughness on nonlinear reflection of weak shock waves.

The authors have recently shown that irregular reflections of spark-generated pressure weak shocks from a smooth rigid surface can be studied using an optical interferometer [Karzova, Lechat, Ollivier, Dragna, Yuldashev, Khokhlova, and Blanc-Benon, J. Acoust. Soc. Am. 145(1), 26-35 (2019)]. The current study extends these results to the reflection from rough surfaces. A Mach-Zehnder interferometer is used to measure pressure waveforms. Simulations are based on the solution of axisymmetric Euler equations. It is shown that roughness causes a decrease of the Mach stem height and the appearance of oscillations in the pressure waveforms. Close to rough surfaces, the pressure was higher compared to the smooth surface.

Irregular reflection of spark-generated shock pulses from a rigid surface: Mach-Zehnder interferometry measurements in air.

The irregular reflection of weak acoustic shock waves, known as the von Neumann reflection, has been observed experimentally and numerically for spherically diverging waves generated by an electric spark source. Two optical measurement methods are used: a Mach-Zehnder interferometer for measuring pressure waveforms and a Schlieren system for visualizing shock fronts. Pressure waveforms are reconstructed from the light phase difference measured by the interferometer using the inverse Abel transform. In numerical simulations, the axisymmetric Euler equations are solved using finite-difference time-domain methods and the spark source is modeled as an instantaneous energy injection with a Gaussian shape. Waveforms and reflection patterns obtained from the simulations are in good agreement with those measured by the interferometer and the Schlieren methods. The Mach stem formation is observed close to the surface for incident pressures within the range of 800 to 4000 Pa. Similarly, as for strong shocks generated by blasts, it is found that for spherical weak shocks the Mach stem length increases with distance following a parabolic law. This study confirms the occurrence of irregular reflections at acoustic pressure levels and demonstrates the benefits of the Mach-Zehnder interferometer method when microphone measurements cannot be applied.

Sound propagation over the ground with a random spatially-varying surface admittance.

Sound propagation over the ground with a random spatially-varying surface admittance is investigated. Starting from the Green's theorem, a Dyson equation is derived for the coherent acoustic pressure. Under the Bourret approximation, an explicit expression is deduced and an effective admittance that depends on the correlation function of the admittance fluctuations is exhibited. An asymptotic expression at long range is then obtained. Influence of the randomness on the amplitude of the reflection coefficient and on the wavenumbers of the surface wave component is analyzed. Afterwards, numerical simulations of the linearized Euler equations are carried out and the coherent pressure obtained by an ensemble-averaging over 200 realizations of the admittance is found to be in good agreement with the analytical solution. In the considered examples of grounds, the mean intensity is shown to be similar to the intensity in the non-random case, except near interferences that are smoothened out due to randomness. It is however exemplified that the intensity fluctuations can be large, especially near destructive interferences.

On the inadvisability of using single parameter impedance models for representing the acoustical properties of ground surfaces.

Although semi-empirical one parameter models are used widely for representing outdoor ground impedance, they are not physically admissible. Even when corrected to satisfy a passivity condition in respect of surface impedance they do not satisfy the condition that the real part of complex density must be greater than zero. Comparison of predictions with frequency-domain data for short range propagation have indicated that physically admissible models provide superior overall agreement. A two parameter variable porosity model yields better agreement for many grassland surfaces and a two parameter version of the slit pore microstructural impedance model yields better agreement with data obtained over low flow resistivity surfaces such as forest floors and gravel. Impedance models and conditions for physical admissibility are summarised. In addition to those examined previously, the slit pore model is shown to be physically admissible. After providing further examples of the better agreement with short range data that can be achieved using two parameter models, it is shown that differences between frequency domain predictions at longer ranges using physically admissible models rather than one parameter models are significantly greater than those resulting from short range spatial variability and comparable with seasonal variability over grassland.

A generalized recursive convolution method for time-domain propagation in porous media.

An efficient numerical method, referred to as the auxiliary differential equation (ADE) method, is proposed to compute convolutions between relaxation functions and acoustic variables arising in sound propagation equations in porous media. For this purpose, the relaxation functions are approximated in the frequency domain by rational functions. The time variation of the convolution is thus governed by first-order differential equations which can be straightforwardly solved. The accuracy of the method is first investigated and compared to that of recursive convolution methods. It is shown that, while recursive convolution methods are first or second-order accurate in time, the ADE method does not introduce any additional error. The ADE method is then applied for outdoor sound propagation using the equations proposed by Wilson et al. in the ground [(2007). Appl. Acoust. 68, 173-200]. A first one-dimensional case is performed showing that only five poles are necessary to accurately approximate the relaxation functions for typical applications. Finally, the ADE method is used to compute sound propagation in a three-dimensional geometry over an absorbing ground. Results obtained with Wilson's equations are compared to those obtained with Zwikker and Kosten's equations and with an impedance surface for different flow resistivities.

Application of the Fourier pseudospectral time-domain method in orthogonal curvilinear coordinates for near-rigid moderately curved surfaces.

The Fourier pseudospectral time-domain method is an efficient wave-based method to model sound propagation in inhomogeneous media. One of the limitations of the method for atmospheric sound propagation purposes is its restriction to a Cartesian grid, confining it to staircase-like geometries. A transform from the physical coordinate system to the curvilinear coordinate system has been applied to solve more arbitrary geometries. For applicability of this method near the boundaries, the acoustic velocity variables are solved for their curvilinear components. The performance of the curvilinear Fourier pseudospectral method is investigated in free field and for outdoor sound propagation over an impedance strip for various types of shapes. Accuracy is shown to be related to the maximum grid stretching ratio and deformation of the boundary shape and computational efficiency is reduced relative to the smallest grid cell in the physical domain. The applicability of the curvilinear Fourier pseudospectral time-domain method is demonstrated by investigating the effect of sound propagation over a hill in a nocturnal boundary layer. With the proposed method, accurate and efficient results for sound propagation over smoothly varying ground surfaces with high impedances can be obtained.

Impulse propagation over a complex site: a comparison of experimental results and numerical predictions.

Results from outdoor acoustic measurements performed in a railway site near Reims in France in May 2010 are compared to those obtained from a finite-difference time-domain solver of the linearized Euler equations. During the experiments, the ground profile and the different ground surface impedances were determined. Meteorological measurements were also performed to deduce mean vertical profiles of wind and temperature. An alarm pistol was used as a source of impulse signals and three microphones were located along a propagation path. The various measured parameters are introduced as input data into the numerical solver. In the frequency domain, the numerical results are in good accordance with the measurements up to a frequency of 2 kHz. In the time domain, except a time shift, the predicted waveforms match the measured waveforms with a close agreement.

Time-domain solver in curvilinear coordinates for outdoor sound propagation over complex terrain.

The current work aims at developing a linearized Euler equations solver in curvilinear coordinates to account for the effects of topography on sound propagation. In applications for transportation noise, the propagation environment as well as the description of acoustic sources is complex, and time-domain methods have proved their capability to deal with both atmospheric and ground effects. First, equations in curvilinear coordinates are examined. Then time-domain boundary conditions initially proposed for a Cartesian coordinate system are implemented in the curvilinear solver. Two test cases dealing with acoustic scattering by an impedance cylinder in a two-dimensional geometry and by an impedance sphere in a three-dimensional geometry are considered to validate the boundary conditions. Accurate solutions are obtained for both rigid and impedance surfaces. Finally, the solver is used to examine a typical outdoor sound propagation problem. It is shown that it is well-suited to study coupled effects of topography, mixed impedance ground and meteorological conditions.