Discussion of sound propagation through the turbulent Martian atmosphere and implications for inference of turbulence spectra.

Chide et al. [J. Acoust. Soc. Am. 155, 420-435 (2024)] provide a first attempt to infer the spectrum of temperature fluctuations on Mars from experimental data on the variances of travel-time and log-amplitude fluctuations recorded by the microphone on board the Perseverance rover. However, the theoretical formulations that were used to interpret the travel-time data have limitations. In addition to explaining those issues, this article also outlines approaches for predicting statistical characteristics of acoustic signals in the Martian atmosphere. In particular, the experimentally observed dependence of the travel-time variance on the propagation range can be attributed to ground-blocking of buoyantly produced turbulent velocity fluctuations and the non-Markov character of phase fluctuations.

An acoustic investigation of the near-surface turbulence on Mars.

The Perseverance rover is carrying out an original acoustic experiment on Mars: the SuperCam microphone records the spherical acoustic waves generated by laser sparks at distances from 2 m to more than 8 m. These N-shaped acoustic waves scatter from the multiple local heterogeneities of the turbulent atmosphere. Therefore, large and random fluctuations of sound travel time and intensity develop as the waves cross the medium. The variances of the travel times and the scintillation index (normalized variance of the sound intensity) are studied within the mathematical formalism of the propagation of spherical acoustic waves through thermal turbulence to infer statistical properties of the Mars atmospheric temperature fluctuation field. The comparison with the theory is made by simplifying assumptions that do not include wind fluctuations and diffraction effects. Two Earth years (about one Martian year) of observations acquired during the maximum convective period (10:00-14:00 Mars local time) show a good agreement between the dataset and the formalism: the travel time variance diverges from the linear Chernov solution exactly where the density of occurrence of the first caustic reaches its maximum. Moreover, on average, waves travel faster than the mean speed of sound due to a fast path effect, which is also observed on Earth. To account for the distribution of turbulent eddies, several power spectra are tested and the best match to observation is obtained with a generalized von Karman spectrum with a shallower slope than the Kolmogorov cascade, ϕ(k)∝(1+k2L2)-4/3. It is associated with an outer scale of turbulence, L, of 11 cm at 2 m above the surface and a standard deviation of 6 K over 9 s for the temperature. These near-surface atmospheric properties are consistent with a weak to moderate wave scattering regime around noon with little saturation. Overall, this study presents an innovative and promising methodology to probe the near-surface atmospheric turbulence on Mars.

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.

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 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.

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.

Evaluation of numerical predictions of sonic boom level variability due to atmospheric turbulence.

A numerical model of full-scale N-wave sonic boom propagation through turbulence is described based on the nonlinear Khokhlov-Zabolotskaya-Kuznetzov (KZK) propagation equation and the most advanced turbulence model used in atmospheric acoustics. This paper presents the first quantitative evaluation of a KZK-based model using data from the recent Sonic Booms in Atmospheric Turbulence measurement campaigns, which produced one of the most extensive databases of full-scale distorted N-waves and concurrent atmospheric parameters. Simulated and measured distributions of the perceived level (PL) metric, which has been used to predict public annoyance due to sonic booms, are compared. For most of the conditions considered, the present model's predictions of the PL variances agree with the measurement to within normal uncertainty, while about half of the mean value predictions agree. The approximate PL distribution measured for high turbulence conditions falls within about 2 dB of the simulated distribution for nearly all probabilities. These favorable results suggest that the KZK-based model is sufficiently accurate for approximating the N-wave PL distribution, and the model may therefore be useful for predicting public reaction to sonic booms in turbulent conditions.

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.

Signatures of microstreaming patterns induced by non-spherically oscillating bubbles.

In this study, we report recent theoretical and experimental developments dealing with the axisymmetric flow surrounding non-spherically oscillating microbubbles. A wide variety of microstreaming patterns is revealed using a theoretical modeling providing exact analytical solutions of the second-order mean flows. The streaming pattern is highly dependent on the modal content of the bubble interface oscillation, including possibly spherical, translational, and nonspherical modes, as well as any combination of these modes. Experimental results on fluid flow induced by a single, non-spherically oscillating bubble in an unbounded fluid are presented and successfully compared to the theoretical predictions.

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.

Statistics of peak overpressure and shock steepness for linear and nonlinear N-wave propagation in a kinematic turbulence.

Linear and nonlinear propagation of high amplitude acoustic pulses through a turbulent layer in air is investigated using a two-dimensional KZK-type (Khokhlov-Zabolotskaya-Kuznetsov) equation. Initial waves are symmetrical N-waves with shock fronts of finite width. A modified von Kármán spectrum model is used to generate random wind velocity fluctuations associated with the turbulence. Physical parameters in simulations correspond to previous laboratory scale experiments where N-waves with 1.4 cm wavelength propagated through a turbulence layer with the outer scale of about 16 cm. Mean value and standard deviation of peak overpressure and shock steepness, as well as cumulative probabilities to observe amplified peak overpressure and shock steepness, are analyzed. Nonlinear propagation effects are shown to enhance pressure level in random foci for moderate initial amplitudes of N-waves thus increasing the probability to observe highly peaked waveforms. Saturation of the pressure level is observed for stronger nonlinear effects. It is shown that in the linear propagation regime, the turbulence mainly leads to the smearing of shock fronts, thus decreasing the probability to observe high values of steepness, whereas nonlinear effects dramatically increase the probability to observe steep shocks.

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.

Mach stem formation in reflection and focusing of weak shock acoustic pulses.

The aim of this study is to show the evidence of Mach stem formation for very weak shock waves with acoustic Mach numbers on the order of 10(-3) to 10(-2). Two representative cases are considered: reflection of shock pulses from a rigid surface and focusing of nonlinear acoustic beams. Reflection experiments are performed in air using spark-generated shock pulses. Shock fronts are visualized using a schlieren system. Both regular and irregular types of reflection are observed. Numerical simulations are performed to demonstrate the Mach stem formation in the focal region of periodic and pulsed nonlinear beams in water.

Mach-Zehnder interferometry method for acoustic shock wave measurements in air and broadband calibration of microphones.

A Mach-Zehnder interferometer is used to measure spherically diverging N-waves in homogeneous air. An electrical spark source is used to generate high-amplitude (1800 Pa at 15 cm from the source) and short duration (50 µs) N-waves. Pressure waveforms are reconstructed from optical phase signals using an Abel-type inversion. It is shown that the interferometric method allows one to reach 0.4 µs of time resolution, which is 6 times better than the time resolution of a 1/8-in. condenser microphone (2.5 µs). Numerical modeling is used to validate the waveform reconstruction method. The waveform reconstruction method provides an error of less than 2% with respect to amplitude in the given experimental conditions. Optical measurement is used as a reference to calibrate a 1/8-in. condenser microphone. The frequency response function of the microphone is obtained by comparing the spectra of the waveforms resulting from optical and acoustical measurements. The optically measured pressure waveforms filtered with the microphone frequency response are in good agreement with the microphone output voltage. Therefore, an optical measurement method based on the Mach-Zehnder interferometer is a reliable tool to accurately characterize evolution of weak shock waves in air and to calibrate broadband acoustical microphones.

Characterization of spark-generated N-waves in air using an optical schlieren method.

Accurate measurement of high-amplitude, broadband shock pulses in air is an important part of laboratory-scale experiments in atmospheric acoustics. Although various methods have been developed, specific drawbacks still exist and need to be addressed. Here, a schlieren optical method was used to reconstruct the pressure signatures of nonlinear spherically diverging short acoustic pulses generated using an electric spark source (2.5 kPa, 33 µs at 10 cm from the source) in homogeneous air. A high-speed camera was used to capture light rays deflected by refractive index inhomogeneities, caused by the acoustic wave. Pressure waveforms were reconstructed from the light intensity patterns in the recorded images using an Abel-type inversion method. Absolute pressure levels were determined by analyzing at different propagation distances the duration of the compression phase of pulses, which changed due to nonlinear propagation effects. Numerical modeling base on the generalized Burgers equation was used to evaluate the smearing of the waveform caused by finite exposure time of the high-speed camera and corresponding limitations in resolution of the schlieren technique. The proposed method allows the study of the evolution of spark-generated shock waves in air starting from the very short distances from the spark, 30 mm, up to 600 mm.

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.

Laboratory-scale experiment to study nonlinear N-wave distortion by thermal turbulence.

The nonlinear propagation of spark-generated N-waves through thermal turbulence is experimentally studied at the laboratory scale under well-controlled conditions. A grid of electrical resistors was used to generate the turbulent field, well described by a modified von Kármán model. A spark source was used to generate high-amplitude (~1500 Pa) and short duration (~50 µs) N-waves. Thousands of waveforms were acquired at distances from 250 to 1750 mm from the source (~15 to 100 wavelengths). The mean values and the probability densities of the peak pressure, the deviation angle, and the rise time of the pressure wave were obtained as functions of propagation distance through turbulence. The peak pressure distributions were described using a generalized gamma distribution, whose coefficients depend on the propagation distance. A line array of microphones was used to analyze the effect of turbulence on the propagation direction. The angle of deviation induced by turbulence was found to be smaller than 15°, which validates the use of the parabolic equation method to model this kind of experiment. The transverse size of the focus regions was estimated to be on the order of the acoustic wavelength for propagation distances longer than 50 wavelengths.