Interference of the direct and ground-reflected waves in the atmosphere with volumetric scatteringa).

The interference of the direct and ground-reflected sound waves is significantly affected by volumetric scattering in the atmosphere, such as scattering by turbulence and forest. In the present article, the existing theory describing this interference is generalized to three somewhat independent but equally important cases. First, the attenuation of the direct and ground-reflected waves caused by backscattering is addressed. Second, the existing theory is extended for statistically quasi-homogeneous turbulence in which the variances and length scales of the temperature and wind velocity fluctuations depend on the height above the ground. Third, the existing theory, which was previously formulated only for near-horizontal sound propagation, is generalized to slanted sound propagation as pertinent to elevated sound sources. Numerical results for slanted propagation demonstrate that atmospheric turbulence can significantly increase the sound pressure level at the interference minima. The extended theory of the interference of the direct and ground-reflected waves in the atmosphere with volumetric scattering is important for practical applications, such as auralization of flying aircraft and sound propagation in a forest, and can be adapted to radio wave propagation.

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.

Influence of ground blocking on the acoustic phase variance in a turbulent atmosphere.

Sound propagation through atmospheric turbulence is important in many applications such as localization of low flying aircraft, sonic boom disturbances, and auralization of aircraft during takeoff and landing. This article extends an isotropic turbulence model in the atmospheric boundary layer to account for ground blocking of buoyancy-produced velocity fluctuations. The extended, anisotropic turbulence model is needed to correctly predict the effect of the largest velocity eddies on the statistical characteristics of sound signals. This model and geometrical acoustics are then employed to derive a closed-form expression for the variance of the phase fluctuations of a spherical sound wave for vertical and slanted propagation, without the use of the Markov approximation. A numerical analysis of this expression indicates significant anisotropy of the phase variance due to the buoyancy-produced velocity fluctuations with ground blocking such that it decreases in the vertical direction and increases in the near-horizontal directions. The newly formulated phase variance is compared with data from an outdoor experiment on vertical and slanted sound propagation. By accounting for ground blocking, much better agreement is obtained between the theoretical predictions and experimental data.

Equations for finite-difference, time-domain simulation of sound propagation in moving media with arbitrary Mach numbers.

Finite-difference time-domain (FDTD) techniques for sound propagation have become increasingly popular. In moving media, such as the atmosphere, starting equations for FDTD calculations are often limited to low Mach numbers, which may result in significant errors. In this article, two coupled equations for the sound pressure and acoustic particle velocity are derived from the linearized fluid dynamic equations. These coupled equations are valid for arbitrary Mach numbers, in the high-frequency approximation, and can be used in FDTD calculations and other methods for sound propagation in moving media. For low Mach numbers, the equations derived are valid for arbitrary frequencies and are consistent with equations from the literature.

Phase-modulated Rice model for statistical distributions of complex signals.

The basic Rice model is commonly used to describe complex signal statistics from randomly scattered waves. It correctly describes weak (Born) scattering, as well as fully saturated scattering, and smoothly interpolates between these extremes. However, the basic Rice model is unsuitable for situations involving scattering by random inhomogeneities spanning a broad range of spatial scales, as commonly occurs for sound scattering by turbulence in the atmospheric boundary layer and other scenarios. In such scenarios, the phase variations are often considerably stronger than those predicted by the basic Rice model. Therefore, the basic Rice model is extended to include a random modulation in the signal phase, which is attributable to the influence of the largest, most energetic inhomogeneities in the propagation medium. Various joint and marginal distributions for the complex signal statistics are derived to incorporate the phase-modulation effect. Approximations of the phase-modulated Rice model involving the Nakagami distribution for amplitude, and the wrapped normal and von Mises distributions for phase, are also developed and analyzed. The phase-modulated Rice model and various approximations are shown to greatly improve agreement with simulated data for sound propagation in the near-ground atmosphere.

Signal power distributions for simulated outdoor sound propagation in varying refractive conditions.

Probability distributions of acoustic signals propagating through the near-ground atmosphere are simulated by the parabolic equation method. The simulations involve propagation at four angles relative to the mean wind, with frequencies of 100, 200, 400, and 800 Hz. The environmental representation includes realistic atmospheric refractive profiles, turbulence, and ground interactions; cases are considered with and without parametric uncertainties in the wind velocity and surface heat flux. The simulated signals are found to span a broad range of scintillation indices, from near zero to exceeding ten. In the absence of uncertainties, the signal power (or intensity) is fit well by a two-parameter gamma distribution, regardless of the frequency and refractive conditions. When the uncertainties are included, three-parameter distributions, namely, the compound gamma or generalized gamma, are needed for a good fit to the simulation data. The compound gamma distribution appears preferable because its parameters have a straight forward interpretation related to the saturation and modulation of the signal by uncertainties.

Ray tracing in a stratified moving atmosphere: Azimuthal deviation in sound propagation.

Ray tracing based on a closed-form expression for the sound propagation path in a stratified moving medium such as a stratified atmosphere is suggested. The approach involves a one-dimensional (1D) integration over the vertical coordinate and can complement existing ray tracing codes based on a set of differential equations. It can also be used as a benchmark solution, because 1D integration is usually computationally simpler than these codes. Using this approach, a closed-form expression for the azimuthal deviation between the true and apparent source location is obtained. The result enables estimation of the azimuthal deviation given a crosswind or remote sensing of the crosswind provided the azimuthal deviation is measured. Based on these formulations, sound propagation in the atmospheric surface layer characterized by the friction velocity and surface heat flux is studied.

Numerical modeling of mesoscale infrasound propagation in the Arctic.

The impacts of characteristic weather events and seasonal patterns on infrasound propagation in the Arctic region are simulated numerically. The methodology utilizes wide-angle parabolic equation methods for a windy atmosphere with inputs provided by radiosonde observations and a high-resolution reanalysis of Arctic weather. The calculations involve horizontal distances up to 200 km for which interactions with the troposphere and lower stratosphere dominate. Among the events examined are two sudden stratospheric warmings, which are found to weaken upward refraction by temperature gradients while creating strongly asymmetric refraction from disturbances to the circumpolar winds. Also examined are polar low events, which are found to enhance negative temperature gradients in the troposphere and thus lead to strong upward refraction. Smaller-scale and topographically driven phenomena, such as low-level jets, katabatic winds, and surface-based temperature inversions, are found to create frequent surface-based ducting out to 100 km. The simulations suggest that horizontal variations in the atmospheric profiles, in response to changing topography and surface property transitions, such as ice boundaries, play an important role in the propagation.

Vertical and slanted sound propagation in the near-ground atmosphere: Coherence and distributions.

Atmospheric turbulence causes acoustic signals to fluctuate and diminishes their coherence. These phenomena are important in applications such as source localization and sonic boom propagation. This article provides formulations for the spatial, cross-frequency, and temporal coherences of narrowband acoustic signals propagating over vertical and slanted paths in the atmosphere. Formulations for single- and two-point distributions of acoustic signals are also overviewed. The theoretical formulations are compared with data from a comprehensive sound propagation experiment carried out in 2018 at the National Wind Technology Center (Boulder, CO). The theories for sound propagation in a turbulent atmosphere, when combined with turbulence models incorporating shear and buoyancy instabilities, correctly predict the measured spatial coherence, which is primarily affected by small-scale isotropic turbulence. For relatively small coherence times, this approach also correctly predicts the temporal coherence. However, the approach underpredicts the cross-frequency coherence and temporal coherence for relatively large coherence times, which are affected by large-scale anisotropic buoyancy-driven velocity fluctuations. For different regimes ranging from unsaturated to fully saturated scattering, the measured distributions agree well with the theoretical predictions.

Vertical and slanted sound propagation in the near-ground atmosphere: Amplitude and phase fluctuations.

Sound propagation along vertical and slanted paths through the near-ground atmosphere impacts detection and localization of low-altitude sound sources, such as small unmanned aerial vehicles, from ground-based microphone arrays. This article experimentally investigates the amplitude and phase fluctuations of acoustic signals propagating along such paths. The experiment involved nine microphones on three horizontal booms mounted at different heights to a 135-m meteorological tower at the National Wind Technology Center (Boulder, CO). A ground-based loudspeaker was placed at the base of the tower for vertical propagation or 56 m from the base of the tower for slanted propagation. Phasor scatterplots qualitatively characterize the amplitude and phase fluctuations of the received signals during different meteorological regimes. The measurements are also compared to a theory describing the log-amplitude and phase variances based on the spectrum of shear and buoyancy driven turbulence near the ground. Generally, the theory correctly predicts the measured log-amplitude variances, which are affected primarily by small-scale, isotropic turbulent eddies. However, the theory overpredicts the measured phase variances, which are affected primarily by large-scale, anisotropic, buoyantly driven eddies. Ground blocking of these large eddies likely explains the overprediction.

Distribution of the two-point product of complex amplitudes in the fully saturated scattering regime.

This Letter considers probability density functions (pdfs) involving products of the complex amplitudes observed at two points (which may, in general, involve separations in space, time, or frequency) in conditions of fully saturated scattering. First, the pdf is derived for the product of the complex amplitude at one point with the conjugate of the complex amplitude at another point. It is shown that the real and imaginary parts of this product each have a variance gamma pdf. Second, expressions are derived for several joint pdfs involving complex amplitude products and powers at two points.

Wave and extra-wide-angle parabolic equations for sound propagation in a moving atmosphere.

The narrow-angle parabolic equation (NAPE) with the effective sound speed approximation (ESSA) is widely used for sound and infrasound propagation in a moving medium such as the atmosphere. However, it is valid only for angles less than 20° with respect to the nominal propagation direction. In this paper, the wave equation and extra-wide-angle parabolic equation (EWAPE) for high-frequency (short-wavelength) sound waves in a moving medium with arbitrary Mach numbers are derived without the ESSA. For relatively smooth variations in the medium velocity, the EWAPE is valid for propagation angles up to 90°. Using the Padé (n,n) series expansion and narrow-angle approximation, the EWAPE is reduced to the wide-angle parabolic equation (WAPE) and NAPE. Versions of these equations are then formulated for low Mach numbers, which is the case that is usually considered in the literature. The phase errors pertinent to the equations considered are studied. It is shown that the equations for low Mach numbers and the WAPE with the ESSA are applicable only under rather restrictive conditions on the medium velocity. An effective numerical implementation of the WAPE for arbitrary Mach numbers in the Padé (1,1) approximation is developed and applied to sound propagation in the atmosphere.

Theory for spectral broadening of narrowband signals in the atmosphere and experiment with an acoustic source onboard an unmanned aerial vehicle.

Narrowband acoustic signals propagating through inhomogeneous moving media, such as atmospheric turbulence, exhibit spectral broadening. This phenomenon has been previously studied with regard to acoustic remote sensing of the atmosphere with sodars and tonal noise of turbofan aircraft engines. In this paper, spectral broadening is studied for line-of-sight sound propagation in a turbulent atmosphere. The averaged power spectral density of a narrowband signal is calculated from a Fourier transform of the temporal mutual coherence function of the sound field. Spectral broadening is characterized by the width of the spectral density. The spectral width is calculated for the Kolmogorov spectra of temperature and wind velocity fluctuations for both horizontal and slanted propagation in the atmosphere. In the latter case, the height-dependence of the wind speed and turbulence parameters is addressed. The spectral width is studied numerically for different meteorological regimes of the atmospheric boundary layer characterized by the friction velocity and the surface sensible heat flux. Theoretical results are compared to experimental data for propagation from an acoustic multi-tone source onboard an unmanned aerial vehicle to a microphone array located on the ground.

Non-Markov character of the phase fluctuations for sound propagation over relatively small ranges in the turbulent atmosphere.

The Markov approximation significantly simplifies formulations for the statistical moments of a wave propagating in a random medium. For the phase fluctuations, the Markov approximation is expected to be valid if the propagation range is much greater than the scale of largest inhomogeneities in a medium. In the atmospheric boundary layer, this scale can be several hundred meters, indicating that the Markov approximation might be inapplicable for relatively small ranges. In this paper, using geometrical acoustics, the correlation function and variance of the phase fluctuations of a plane sound wave are calculated without the Markov approximation and compared to previous results based on this approximation. The mean sound field and the spatial mutual coherence function (MCF) are also analyzed by expressing them in terms of the phase fluctuations. It is shown that for ranges smaller than the scale of largest inhomogeneities, the variance of the phase fluctuations is significantly smaller than that found with the Markov approximation. For large ranges, the relative difference between the two results tends to zero, while the absolute difference remains constant and can be much greater than unity. For the MCF, the Markov approximation is valid for both small and large ranges.

Extra-wide-angle parabolic equations in motionless and moving media.

Wide-angle parabolic equations (WAPEs) play an important role in physics. They are derived by an expansion of a square-root pseudo-differential operator in one-way wave equations, and then solved by finite-difference techniques. In the present paper, a different approach is suggested. The starting point is an extra-wide-angle parabolic equation (EWAPE) valid for small variations of the refractive index of a medium. This equation is written in an integral form, solved by a perturbation technique, and transformed to the spectral domain. The resulting split-step spectral algorithm for the EWAPE accounts for the propagation angles up to 90° with respect to the nominal direction. This EWAPE is also generalized to large variations in the refractive index. It is shown that WAPEs known in the literature are particular cases of the two EWAPEs. This provides an alternative derivation of the WAPEs, enables a better understanding of the underlying physics and ranges of their applicability, and opens an opportunity for innovative algorithms. Sound propagation in both motionless and moving media is considered. The split-step spectral algorithm is particularly useful in the latter case since complicated partial derivatives of the sound pressure and medium velocity reduce to wave vectors (essentially, propagation angles) in the spectral domain.

An experimental study of the atmospheric-driven variability of impulse sounds.

Propagating impulse sounds are sensitive to the varying near-surface atmosphere. This study reports on an experimental assessment of this sensitivity under well-controlled outdoor conditions. The experiment, conducted over a flat terrain, features 14 synchronous acoustic sensors at ranges up to 450 m from reproducible, transient sources. It scanned over the upwind, crosswind, and downwind propagations, and also documents the temporal and spatial coherences of the acoustic field. Concurrent atmospheric measurements documented the near-surface, essentially wind-driven atmosphere, and included turbulence monitoring. The analysis reveals how the environmental propagation processes combine to form the large variety of recorded signatures. The deterministic versus stochastic variations of the signatures are distinguished, and both are shown to affect the time of arrival (wander) and the shape (spread) of the pulses. The study also discusses the potential impacts of these variations on acoustic sensing of transient signals like gun shots and explosions.

Correspondence between sound propagation in discrete and continuous random media with application to forest acoustics.

Although sound propagation in a forest is important in several applications, there are currently no rigorous yet computationally tractable prediction methods. Due to the complexity of sound scattering in a forest, it is natural to formulate the problem stochastically. In this paper, it is demonstrated that the equations for the statistical moments of the sound field propagating in a forest have the same form as those for sound propagation in a turbulent atmosphere if the scattering properties of the two media are expressed in terms of the differential scattering and total cross sections. Using the existing theories for sound propagation in a turbulent atmosphere, this analogy enables the derivation of several results for predicting forest acoustics. In particular, the second-moment parabolic equation is formulated for the spatial correlation function of the sound field propagating above an impedance ground in a forest with micrometeorology. Effective numerical techniques for solving this equation have been developed in atmospheric acoustics. In another example, formulas are obtained that describe the effect of a forest on the interference between the direct and ground-reflected waves. The formulated correspondence between wave propagation in discrete and continuous random media can also be used in other fields of physics.

Acoustic pulse propagation in forests.

The propagation of acoustic pulses through a forest is considered. Multiple-scattering effects are accounted for by using the energy-based radiative transfer theory under a modified Born approximation, resulting in an expression for the diffuse intensity as a function of time and dominant frequency. While this expression is a complicated set of three integrals, certain practical approximations enable analytic evaluation of one, two, or even all three integrals. Any remaining integrals may be numerically calculated. The simple case of an impulse in an infinite homogeneous forest of diffuse scatterers is first considered, and then the effects of successively including non-diffuse scatterers, ground reflections in a forest of finite height, and finally a realistic forest model are analyzed with an emphasis on long-time decay and reverberation times. These theoretical findings are then compared with experimental results.