A characteristic nonlinear distortion length for broadband Gaussian noise.

The nonlinear evolution of high-amplitude broadband noise is important to the psychoacoustic perception, usually annoyance, of high-speed jet noise. One method to characterize the nonlinear evolution of such noise is to consider a characteristic nonlinear waveform distortion length for the signal. A common length scale for this analysis is the shock formation distance of an initially sinusoidal signal. However, application of this length scale to broadband noise, even with the amplitude and source frequency replaced with characteristic values, may lead to underestimates of the overall nonlinear waveform distortion of the noise as indicated by the skewness of the time derivative of the acoustic pressure (or derivative skewness). This paper provides an alternative length scale derived directly from the evolution of the derivative skewness of Gaussian noise that may be more appropriate when analyzing the nonlinear evolution of broadband noise signals. This Gaussian-based length scale is shown to be a useful metric for its relative consistency and its physical interpretation. Various analytical predictions of the evolution of the derivative skewness for an ensemble of numerical simulations of noise propagation are used to highlight various aspects of this new length scale definition.

Weak-form homogenization of two and three-dimensional fluid acoustical systems.

A one-dimensional weak-form homogenization method [Muhlestein, J. Acoust. Soc. Am. 147(5), 3584-3593 (2020)] is extended to two and three-dimensional for quasi-static fluid systems. This homogenization approach uses a local multiple-scales approximation to estimate the acoustical fields within a representative volume element, substitutes these approximations into a weak formulation of the mechanics, and then globally homogenizes the system by averaging the integrand of the weak-form integral. An important consequence of including more spatial dimensions is that the local particle velocity does not approach a uniform macroscopic particle velocity. Instead, the effective material properties are used to describe the behavior of the mean particle velocity. A localization tensor may be used to convert from the mean particle velocity to the local particle velocity. The generalized homogenization method is then applied to two special cases. The first case is stratified media, chosen because it has an exact analytical solution. The second case is a cubic lattice of spheres, which has a benchmark solution to compare with. This second case utilizes finite element software to provide estimates of the effective mass density. Finally, three further generalizations to the homogenization method, including extension to finite frequency values, complex media, and elasticity, are briefly discussed.

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.

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.

Geometric-acoustics analysis of singly scattered, nonlinearly evolving waves by circular cylinders.

Geometric acoustics, or acoustic ray theory, is used to analyze the scattering of high-amplitude acoustic waves incident upon rigid circular cylinders. Theoretical predictions of the nonlinear evolution of the scattered wave field are provided, as well as measures of the importance of accounting for nonlinearity. An analysis of scattering by many cylinders is also provided, though the effects of multiple scattering are not considered. Provided the characteristic nonlinear distortion length is much larger than a cylinder radius, the nonlinear evolution of the incident wave is shown to be of much greater importance to the overall evolution than the nonlinear evolution of the individual scattered waves.

Effective acoustic metamaterial homogenization based on Hamilton's principle with a multiple scales approximation.

This paper derives and demonstrates a one-dimensional acoustic metamaterial homogenization method. The homogenization method uses a multiple-scales approximation with Hamilton's principle, a weak-form representation of the dynamic equation. While the multiple-scales approximation makes the predicted effective material properties of this method inexact, the method is shown to be highly versatile. Analytical and numerical examples are given showing the ability of the homogenization method to account for viscosity and finite-amplitude effects.

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.

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.

Acoustic scattering from a fluid cylinder with Willis constitutive properties.

A material that exhibits Willis coupling has constitutive equations that couple the pressure-strain and momentum-velocity relationships. This coupling arises from subwavelength asymmetry and non-locality in heterogeneous media. This paper considers the problem of the scattering of a plane wave by a cylinder exhibiting Willis coupling using both analytical and numerical approaches. First, a perturbation method is used to describe the influence of Willis coupling on the scattered field to a first-order approximation. A higher order analysis of the scattering based on generalized impedances is then derived. Finally, a finite-element method-based numerical scheme for calculating the scattered field is presented. These three analyses are compared and show strong agreement for low to moderate levels of Willis coupling.

Effective wavenumbers for sound scattering by trunks, branches, and the canopy in a forest.

Sound propagation in a forest is often represented as propagation in free space with an effective complex wavenumber, which accounts for scattering and absorption. In this paper, the effective wavenumbers due to sound scattering by trunks, large branches, and the canopy are determined and analyzed based on three-dimensional multiple scattering theory. Trunks and branches are modeled as vertical and slanted finite cylinders, while the canopy is modeled by diffuse scatterers. The results are compared with two-dimensional effective wavenumbers previously used in the literature, which were obtained by approximating the trunk layer as infinite vertical cylinders.

Experimental evidence of Willis coupling in a one-dimensional effective material element.

The primary objective of acoustic metamaterial research is to design subwavelength systems that behave as effective materials with novel acoustical properties. One such property couples the stress-strain and the momentum-velocity relations. This response is analogous to bianisotropy in electromagnetism, is absent from common materials, and is often referred to as Willis coupling after J.R., Willis, who first described it in the context of the dynamic response of heterogeneous elastic media. This work presents two principal results: first, experimental and theoretical demonstrations, illustrating that Willis properties are required to obtain physically meaningful effective material properties resulting solely from local behaviour of an asymmetric one-dimensional isolated element and, second, an experimental procedure to extract the effective material properties from a one-dimensional isolated element. The measured material properties are in very good agreement with theoretical predictions and thus provide improved understanding of the physical mechanisms leading to Willis coupling in acoustic metamaterials.

Radiative transfer formulation for forest acoustics.

The radiative transfer equation (RTE) is outlined and then applied to forest acoustics. The RTE is an integro-differential equation for the radiance, which is the angular Fourier transform of the spatial correlation function of the sound field. It correctly accounts for propagation phenomena such as multiple scattering, absorption, and the transformation of the coherent sound field into the incoherent field. In this formulation, acoustical properties of a forest are described by the total cross sections and differential scattering cross sections of different scatterers. In this paper, the four-layer forest model is used with the following distinct layers: ground, trunk layer, canopy layer, and open air. The trunk layer is modeled with finite vertical cylinders, while the canopy layer is modeled with diffuse scatterers. The total and differential scattering cross sections in these layers are calculated. The boundary condition for the radiance at the interface between the ground and trunk layers is formulated. Using a modified Born approximation, the RTE is solved for a plane sound wave normally incident on the edge of a forest. The mean intensities transmitted and backscattered from a stand of trees are calculated and analyzed.

Reciprocity, passivity and causality in Willis materials.

Materials that require coupling between the stress-strain and momentum-velocity constitutive relations were first proposed by Willis (Willis 1981 Wave Motion3, 1-11. (doi:10.1016/0165-2125(81)90008-1)) and are now known as elastic materials of the Willis type, or simply Willis materials. As coupling between these two constitutive equations is a generalization of standard elastodynamic theory, restrictions on the physically admissible material properties for Willis materials should be similarly generalized. This paper derives restrictions imposed on the material properties of Willis materials when they are assumed to be reciprocal, passive and causal. Considerations of causality and low-order dispersion suggest an alternative formulation of the standard Willis equations. The alternative formulation provides improved insight into the subwavelength physical behaviour leading to Willis material properties and is amenable to time-domain analyses. Finally, the results initially obtained for a generally elastic material are specialized to the acoustic limit.

A micromechanical approach for homogenization of elastic metamaterials with dynamic microstructure.

An approximate homogenization technique is presented for generally anisotropic elastic metamaterials consisting of an elastic host material containing randomly distributed heterogeneities displaying frequency-dependent material properties. The dynamic response may arise from relaxation processes such as viscoelasticity or from dynamic microstructure. A Green's function approach is used to model elastic inhomogeneities embedded within a uniform elastic matrix as force sources that are excited by a time-varying, spatially uniform displacement field. Assuming dynamic subwavelength inhomogeneities only interact through their volume-averaged fields implies the macroscopic stress and momentum density fields are functions of both the microscopic strain and velocity fields, and may be related to the macroscopic strain and velocity fields through localization tensors. The macroscopic and microscopic fields are combined to yield a homogenization scheme that predicts the local effective stiffness, density and coupling tensors for an effective Willis-type constitutive equation. It is shown that when internal degrees of freedom of the inhomogeneities are present, Willis-type coupling becomes necessary on the macroscale. To demonstrate the utility of the homogenization technique, the effective properties of an isotropic elastic matrix material containing isotropic and anisotropic spherical inhomogeneities, isotropic spheroidal inhomogeneities and isotropic dynamic spherical inhomogeneities are presented and discussed.

Evolution of the derivative skewness for nonlinearly propagating waves.

The skewness of the first time derivative of a pressure waveform, or derivative skewness, has been used previously to describe the presence of shock-like content in jet and rocket noise. Despite its use, a quantitative understanding of derivative skewness values has been lacking. In this paper, the derivative skewness for nonlinearly propagating waves is investigated using analytical, numerical, and experimental methods. Analytical expressions for the derivative skewness of an initially sinusoidal plane wave are developed and, along with numerical data, are used to describe its behavior in the preshock, sawtooth, and old-age regions. Analyses of common measurement issues show that the derivative skewness is relatively sensitive to the effects of a smaller sampling rate, but less sensitive to the presence of additive noise. In addition, the derivative skewness of nonlinearly propagating noise is found to reach greater values over a shorter length scale relative to sinusoidal signals. A minimum sampling rate is recommended for sinusoidal signals to accurately estimate derivative skewness values up to five, which serves as an approximate threshold indicating significant shock formation.

Evolution of the temporal slope density function for waves propagating according to the inviscid Burgers equation.

An exact formulation for the evolution of the probability density function of the time derivative of a waveform (slope density) propagating according to the one-dimensional inviscid Burgers equation is given. The formulation relies on the implicit Earnshaw solution and therefore is only valid prior to shock formation. As explicit examples, the slope density evolution of an initially sinusoidal plane wave, initially Gaussian-distributed planar noise, and an initially triangular wave are presented. The triangular wave is used to examine weak-shock limits without violating the theoretical assumptions. It is also shown that the moments of the slope density function as a function of distance may be written as an expansion in terms of the moments of the source slope density function. From this expansion, approximate expressions are presented for the above cases as well as a specific non-Gaussian noise case intended to mimic features of jet noise. Finally, analytical predictions of the propagation of initially Gaussian-distributed noise are compared favorably with plane-wave tube measurements.

Evolution of the average steepening factor for nonlinearly propagating waves.

Difficulties arise in attempting to discern the effects of nonlinearity in near-field jet-noise measurements due to the complicated source structure of high-velocity jets. This article describes a measure that may be used to help quantify the effects of nonlinearity on waveform propagation. This measure, called the average steepening factor (ASF), is the ratio of the average positive slope in a time waveform to the average negative slope. The ASF is the inverse of the wave steepening factor defined originally by Gallagher [AIAA Paper No. 82-0416 (1982)]. An analytical description of the ASF evolution is given for benchmark cases-initially sinusoidal plane waves propagating through lossless and thermoviscous media. The effects of finite sampling rates and measurement noise on ASF estimation from measured waveforms are discussed. The evolution of initially broadband Gaussian noise and signals propagating in media with realistic absorption are described using numerical and experimental methods. The ASF is found to be relatively sensitive to measurement noise but is a relatively robust measure for limited sampling rates. The ASF is found to increase more slowly for initially Gaussian noise signals than for initially sinusoidal signals of the same level, indicating the average distortion within noise waveforms occur more slowly.