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When waves propagate through a complex medium like the turbulent atmosphere the wave field becomes incoherent and the wave intensity forms a complex speckle pattern. In this paper we study a speckle memory effect in the frequency domain and some of its consequences. This effect means that certain properties of the speckle pattern produced by wave transmission through a randomly scattering medium is preserved when shifting the frequency of the illumination. The speckle memory effect is characterized via a detailed novel analysis of the fourth-order moment of the random paraxial Green's function at four different frequencies. We arrive at a precise characterization of the frequency memory effect and what governs the strength of the memory. As an application we quantify the statistical stability of time-reversal wave refocusing through a randomly scattering medium in the paraxial or beam regime. Time reversal refers to the situation when a transmitted wave field is recorded on a time-reversal mirror then time reversed and sent back into the complex medium. The re-emitted wave field then refocuses at the original source point. We compute the mean of the refocused wave and identify a novel quantitative description of its variance in terms of the radius of the time-reversal mirror, the size of its elements, the source bandwidth, and the statistics of the random medium fluctuations. [ABSTRACT FROM AUTHOR]

The production of broadband frequency spectra from narrowband wave forcing in geophysical flows remains an open problem. Here we consider a related theoretical problem that points to the role of time-dependent vortical flow in producing this effect. Specifically, we apply multi-scale analysis to the transport equation of wave action density in a homogeneous stationary random background flow under the Wentzel–Kramers–Brillouin approximation. We find that, when some time dependence in the mean flow is retained, wave action density diffuses both along and across surfaces of constant frequency in wavenumber–frequency space; this stands in contrast to previous results showing that diffusion occurs only along constant-frequency surfaces when the mean flow is steady. A self-similar random background velocity field is used to show that the magnitude of this frequency diffusion depends non-monotonically on the time scale of variation of the velocity field. Numerical solutions of the ray-tracing equations for rotating shallow water illustrate and confirm our theoretical predictions. Notably, the mean intrinsic wave frequency increases in time, which by wave action conservation implies a concomitant increase of wave energy at the expense of the energy of the background flow. [ABSTRACT FROM AUTHOR]

We use a multiple-scale expansion to average the wave action balance equation over an ensemble of sea-surface velocity fields characteristic of the ocean mesoscale and submesoscale. Assuming that the statistical properties of the flow are stationary and homogeneous, we derive an expression for a diffusivity tensor of surface-wave action density. The small parameter in this expansion is the ratio of surface current speed to gravity wave group speed. For isotropic currents, the action diffusivity is expressed in terms of the kinetic energy spectrum of the flow. A Helmholtz decomposition of the sea-surface currents into solenoidal (vortical) and potential (divergent) components shows that, to leading order, the potential component of the surface velocity field has no effect on the diffusivity of wave action: only the vortical component of the sea-surface velocity results in diffusion of surface-wave action. We validate our analytic results for the action diffusivity by Monte Carlo ray-tracing simulations through an ensemble of stochastic velocity fields. [ABSTRACT FROM AUTHOR]

When waves propagate through a strongly scattering medium the energy is transferred to the incoherent wave part by scattering. The wave intensity then forms a random speckle pattern seemingly without much useful information. However, a number of recent physical experiments show how one can extract useful information from this speckle pattern. Here we present the mathematical analysis that explains the quite stunning performance of such a scheme for speckle imaging. Our analysis is based on the white-noise paraxial model, in which the wave amplitude is described by the Itô{Schrödinger equation. We identify a scaling regime where the scheme works well, which we refer to as the scintillation regime. In this regime the wavelength is smaller than the correlation radius of the medium, which in turn is smaller than the beam radius; moreover, the propagation distance is longest scale. The results presented in this paper conform with the sophisticated physical intuition that has motivated these schemes, but give a more detailed characterization of the performance. The analysis gives a description of (i) the information that can be extracted and with what resolution and (ii) the statistical stability or signal-to-noise ratio with which the information can be extracted. [ABSTRACT FROM AUTHOR]

Novel acoustic liner designs often incorporate new materials with non-uniform impedance distributions. Therefore, new methods are required for their modelling and analysis. In this paper, a theoretical model is developed to investigate the scattering of sound waves from an axially symmetrical flow duct with a semi-infinite, azimuthally non-uniform acoustic lining on the duct wall. More specifically, the incorporation of Fourier series expansions into the Wiener–Hopf method leads to an analytical model with a matrix kernel, which is further factorised by using the pole-removal method to obtain a closed-form solution. A new mathematical method is developed to solve the residues associated with the pole-removal technique. The proposed model has been verified and validated by comparing with corresponding computational results. In addition to shedding light on the possible physical effect of azimuthally non-uniform liners along with an axial hard–soft interface, the current model enhances the theoretical modelling capability for a complicated set-up of practical importance, and can be used to investigate new liner designs for passive noise control in flow ducts. [ABSTRACT FROM AUTHOR]

When waves propagate through a complex or heterogeneous medium the wave field is corrupted by the heterogeneities. Such corruption limits the performance of imaging or communication schemes. One may then ask the question, Is there an optimal way of encoding a signal so as to counteract the corruption by the medium? In the ideal situation the answer is given by time reversal: for a given target or focusing point, in a first step let the target emit a signal and then record the signal transmitted to the source antenna, time reverse this, and use it as the source trace at the source antenna in a second step. This source will give a sharply focused wave at the target location if the source aperture is large enough. Here we address this scheme in the more practical situation with a limited aperture, time-harmonic signal, and finite-sized elements in the source array. Central questions are then the focusing resolution and signal-to-noise ratio at the target, their dependence on the physical parameters, and the capacity to focus selectively in the neighborhood of the target point and therefore to transmit images. Sharp results are presented for these questions. [ABSTRACT FROM AUTHOR]

Linear potential-flow theory is used to study loads imposed on finite line arrays of rigid, bottom-mounted, surface-piercing, vertical cylinders by surface water waves. Perturbations in the cylinder locations are shown to damp the resonant loads experienced by the unperturbed array. A relationship is established between the damping and the phenomenon of Anderson localisation. Specifically, the Rayleigh–Bloch waves responsible for the resonant loads are shown to attenuate along the array when perturbations are introduced, resulting in localisation when the attenuation rate is sufficiently large with respect to the array length. Further, an efficient solution method for line arrays is introduced that captures the Rayleigh–Bloch wave modes supported by unperturbed arrays from the scattering characteristics of an individual cylinder. [ABSTRACT FROM AUTHOR]

We study an inverse problem for the wave equation where localized wave sources in random scattering media are to be determined from time resolved measurements of the waves at an array of receivers. The sources are far from the array, so the measurements are affected by cumulative scattering in the medium, but they are not further than a transport mean free path, which is the length scale characteristic of the onset of wave diffusion that prohibits coherent imaging. The inversion is based on the coherent interferometric (CINT) imaging method, which mitigates the scattering effects by introducing an appropriate smoothing operation in the image formation. This smoothing stabilizes the images statistically, at the expense of their resolution. We complement the CINT method with a convex (l1) optimization in order to improve the source localization and obtain quantitative estimates of the source intensities. We analyze the method in a regime where scattering can be modeled by large random wavefront distortions and quantify the accuracy of the inversion in terms of the spatial separation of individual sources or clusters of sources. The theoretical predictions are demonstrated with numerical simulations. [ABSTRACT FROM AUTHOR]

Waves propagating through heterogeneous media experience scattering that can convert a coherent pulse into small incoherent fluctuations. This may appear as attenuation for the transmitted front pulse. The classic O'Doherty-Anstey theory describes such a transformation for scalar waves in finely layered media. Recent observations for seismic waves in the earth suggest that this theory can explain a significant component of seismic attenuation. An important question to answer is then how the O'Doherty-Anstey theory generalizes to seismic waves when several wave modes, possibly with the same velocity, interact. An important aspect of the O'Doherty-Anstey theory is the statistical stability property, which means that the transmitted front pulse is actually deterministic and depends only on the statistics of the medium but not on the particular medium realization when the medium is modeled as a random process. It is shown in this paper that this property generalizes in the case of elastic waves in a nontrivial way: the energy of the transmitted front pulse, but not the pulse shape itself, is statistically stable. This result is based on a separation of scales technique and a diffusion-approximation theorem that characterize the transmitted front pulse as the solution of a stochastic partial differential equation driven by two Brownian motions. [ABSTRACT FROM AUTHOR]

In the weakly heterogeneous regime of elastic wave propagation through a random medium, transport and diffusion models for the energy densities can be set up. In the isotropic case, the scattering cross sections are explicitly known as a function of the wavenumber and the correlations of the Lamé parameters and density. In this paper, we discuss the precise influence of the correlation structure on the scattering cross sections, mean free paths and diffusion parameter, and separate that influence from that of the correlation length and variance. We also analyze the convergence rates towards the low- and high-frequency ranges. For all analyses, we consider five different correlation structures that allow us to explore a wide range of behaviors. We identify that the controlling factors for the low-frequency behavior are the value of the Power Spectral Density Function (PSDF) and its first non-vanishing derivative at the origin. In the high frequency range, the controlling factor is the third moment of the PSDF (which may be unbounded). [ABSTRACT FROM AUTHOR]

When waves penetrate a medium without coherent reflectors, but with some fine scale medium heterogeneities, the backscattered wave is incoherent without any specific arrival time or the like. In this paper we consider a distributed field of weak microscatterers, like aerosols in the atmosphere, which coexists with microstructured clutter in the medium, like the fluctuations of the index of refraction of the turbulent atmosphere. We analyze the Wigner transform or the angularly resolved intensity profile of the backscattered wave when the incident wave is a beam in the paraxial regime and when the Born approximation is valid for the microscatterers. An enhanced backscattering phenomenon is proved, and the properties of the enhanced backscattering cone (relative amplitude and profile) are shown to depend on the statistical parameters of the microstructure but not on the microscatterers. These results are based on a multiscale analysis of the fourth-order moment of the fundamental solution of the white-noise paraxial wave equation. They pave the way for an estimation method of the statistical parameters of the microstructure from the observation of the enhanced backscattering cone. In our scaling argument we differentiate the two important canonical scaling regimes, which are the scintillation regime and the spot dancing regime. [ABSTRACT FROM AUTHOR]

In this paper the white-noise paraxial wave model is considered. This model describes for instance the propagation of laser beams in the atmosphere in some typical scaling regimes. The closed-form equations for the second- and fourth-order moments of the field are solved in two particular situations. The first situation corresponds to a random medium with a transverse correlation radius smaller than the beam radius. This is the spot-dancing regime: the beam shape spreads out as in a homogeneous medium and its center is randomly shifted according to a Gaussian process whose variance grows like the third power of the propagation distance. The second situation corresponds to a plane-wave initial condition, a small amplitude for the medium fluctuations, and a large propagation distance. This is the scintillation regime: the normalized variance of the intensity converges to one exponentially with the propagation distance, corresponding to strong intensity fluctuations and in agreement with the conjecture that the statistics of the field becomes complex Gaussian. [ABSTRACT FROM AUTHOR]

High frequency waves propagating in highly oscillatory media are often modeled by radiative transfer equations that describes the propagation of the energy density of the waves. When the medium is statistically homogeneous, averaging effects occur in such a way that in the limit of vanishing wavelength, the wave energy density solves a deterministic radiative transfer equation. In this paper, we are interested in the remaining stochasticity of the energy density. More precisely, we wish to understand how such stochasticity depends on the statistics of the random medium and on the initial phase-space structure of the propagating wave packets. The analysis of stochasticity is a formidable task involving complicated analytical calculations. In this paper, we consider the propagation of waves modeled by a scalar Schrödinger equation and limit the interaction of the waves with the underlying structure to second order. We calculate the scintillation function (second statistical moment of the Wigner transform) for such signals, which thus involve fourth-order moments of the random fluctuations, which we assume have Gaussian statistics. Our main result is a detailed analysis of the scintillation function in that setting. This requires the analysis of non-trivial oscillatory integrals, which is carried out in detail. [ABSTRACT FROM AUTHOR]

In this paper the collective oscillations of a bubble cloud in an acoustic field are theoretically analysed with concepts and techniques of condensed matter physics. More specifically, we will calculate the eigenmodes and their excitabilities, eigenfrequencies, densities of states, responses, absorption and participation ratios to better understand the collective dynamics of coupled bubbles and address the question of possible localization of acoustic energy in the bubble cloud. The radial oscillations of the individual bubbles in the acoustic field are described by coupled linearized Rayleigh–Plesset equations. We explore the effects of viscous damping, distance between bubbles, polydispersity, geometric disorder, size of the bubbles and size of the cloud. For large enough clusters, the collective response is often very different from that of a typical mode, as the frequency response of each mode is sufficiently wide that many modes are excited when the cloud is driven by ultrasound. The reason is the strong effect of viscosity on the collective mode response, which is surprising, as viscous damping effects are small for single-bubble oscillations in water. Localization of acoustic energy is only found in the case of substantial bubble size polydispersity or geometric disorder. The lack of localization for a weak disorder is traced back to the long-range 1/r interaction potential between the individual bubbles. The results of the present paper are connected to recent experimental observations of collective bubble oscillations in a two-dimensional bubble cloud, where pronounced edge states and a pronounced low-frequency response had been observed, both consistent with the present theoretical findings. Finally, an outlook to future possible experiments is given. [ABSTRACT FROM PUBLISHER]

This paper concerns the asymptotic structure of the scintillation function in the simplified setting of wave propagation modeled by an Ito-Schrodinger equation. We show that the size of the scintillation function crucially depends on the smoothness of the initial conditions for the wave equation and on the size of the “array of detectors” where the wave fields are measured. In many practical settings, we show that the estimates are optimal and devise an equation for the appropriately rescaled scintillation function. The estimates are based on a careful analysis of Wigner transforms and of linear kinetic equations involving oscillatory integrals. [ABSTRACT FROM AUTHOR]

We derive kinetic models for the correlations and the energy densities of wave fields propagating in random media. These models take the form of radiative transfer and diffusion equations. We use these macroscopic models to address the detection and imaging of small objects buried in highly heterogeneous media. More specifically, we quantify the influence of small objects on (i) the energy density measured at an array of detectors and (ii) the correlation between the wave field measured in the absence of the object and the wave field measured in the presence of the object. We analyze the advantages and disadvantages of such measurements as a function of the level of disorder in the random media. Numerical simulations verify the theoretical predictions. [ABSTRACT FROM AUTHOR]

Abstract: Determining the effects of material spatial randomness on the distance to form shocks from acceleration waves, x ∞, in random media is the objective of the present study. A very general class of random media is modeled by two random fields—the dissipation (μ) and elastic nonlinearity (β). The reason for considering the randomness of said material coefficients is the fact that a wavefront’s length scale is not necessarily greater than the representative volume element—a condition tacitly assumed in deterministic continuum mechanics. There are two entirely new aspects considered in the present study. One is the explicit consideration of μ and β as functions of four more fundamental material properties, and themselves random fields: the instantaneous modulus (G 0), the dissipation coefficient , the instantaneous second-order tangent modulus , the mass density in the reference state (ρ R). The second new facet is the coupling of the four-component random field to the wavefront amplitude α, because as the amplitude grows, the wavefront gets thinner tending to a shock, and thus the material random heterogeneity shows up as a random field with ever stronger fluctuations. In effect, the wavefront is an object which is more appropriately analyzed as a statistical volume element, and therefore to be treated via a stochastic rather than a deterministic dynamical system. [Copyright &y& Elsevier]

This paper analyzes the refocusing properties of time-reversed acoustic waves that propagate in different media during the forward and backward propagation phases. We show how the refocused signal is modified as the medium during backward propagation departs from the medium during forward propagation. The derivation is based on the analysis of the correlation of two fields propagating in different backgrounds. The correlation is described by the Wigner transform of the two fields, which satisfies in the limit of high frequencies a generalized radiative transfer equation. The theory is presented in two and three space dimensions in the transport and diffusive regimes. Numerical experiments show a remarkable agreement with theory. [ABSTRACT FROM AUTHOR]

There are several important parameters, such as coherence time, coherence distance and coherence bandwidth, which describe the characteristics of the signal fluctuations and the channel model. These quantities are determined by two-position, two-frequency and two-time mutual coherence function of the received signals after propagating through random media. This paper discusses these second-order statistical quantities by using an analytic solution to the mutual coherence function for plane wave recently obtained by iteration of an integral equation. As an example, the coherence distance is numerically discussed according to signal frequencies and parameters of ionospheric irregularities. It is found that the coherence distance increases with both outer and inner scale, more quickly with the former, while it decreases quickly with increasing of fluctuations of the electron density. Afterwards, the power impulse response, spectra and delay-Doppler scattering function are also derived and discussed with scaled version of illustration. All of these quantities are of importance and necessary to describing a channel model for transionospheric radio propagation through the structured ionosphere. [Copyright &y& Elsevier]

We consider the stabilization (self-averaging) and destabilization of the energy of waves propagating in random media. Propagation is modeled here by an Itô-Schrödinger equation. The explicit structure of the resulting transport equations for arbitrary statistical moments of the wave field is used to show that wave energy density may be stable in the high frequency regime, in the sense that it depends only on the statistics of the random medium and not on the specific realization. Stability is conditional on having sufficiently smooth initial energy distributions. We show that wave energy is not stable, and instead scintillation is created by the wave dynamics, when the initial energy distribution is sufficiently singular. Application to time reversal of high frequency waves is also considered. [ABSTRACT FROM AUTHOR]

We study the effects of material spatial randomness on the distance to form shocks from acceleration waves, x∞, in random media. We introduce this randomness by taking the material coefficients μ and β — that represent the dissipation and elastic nonlinearity, respectively, in the governing Bernoulli equation — as a stochastic vector process. The focus of our investigation is the resulting stochastic, rather than deterministic as in classical continuum mechanics studies, competition of dissipation and elastic nonlinearity. Quantitative results for x∞ are obtained by the method of moments in special simple cases, and otherwise by the method of maximum entropy. We find that the effect of even very weak random perturbation in μ and β may be very significant on x∞. In particular, the full negative cross-correlation between μ and β results in the strongest scatter of x∞, and hence, in the largest probability of shock formation in a given distance x. [ABSTRACT FROM AUTHOR]

We present a theoretical derivation of acoustic phonon damping in amorphous solids based on the nonaffine response formalism for the viscoelasticity of amorphous solids. The analytical theory takes into account the nonaffine displacements in transverse waves and is able to predict both the ubiquitous low-energy diffusive damping ∼ k 2 , as well as a novel contribution to the Rayleigh damping ∼ k 4 at higher wavevectors and the crossover between the two regimes observed experimentally. The coefficient of the diffusive term is proportional to the microscopic viscous (Langevin-type) damping in particle motion (which arises from anharmonicity), and to the nonaffine correction to the static shear modulus, whereas the Rayleigh damping emerges in the limit of low anharmonicity, consistent with previous observations and macroscopic models. Importantly, the k 4 Rayleigh contribution derived here does not arise from harmonic disorder or elastic heterogeneity effects and it is the dominant mechanism for sound attenuation in amorphous solids as recently suggested by molecular simulations., (© 2022 IOP Publishing Ltd.)

We derive radiative transport equations for solutions of a Schrödinger equation in a periodic structure with small random inhomogeneities. We use systematically the Wigner transform and the Bloch wave expansion. The streaming part of the radiative transport equations is determined entirely by the Bloch spectrum, and the scattering part by the random fluctuations. [ABSTRACT FROM AUTHOR]

We study the simultaneous paraxial and white-noise limit of the Helmholtz equation in randomly layered media where the refractive index fluctuations are in the direction of propagation. We consider the regime in which the wavelength is of the same order as the correlation length of the random fluctuations of the refractive index. We show that this simultaneous limit can be taken in this regime by introducing into the equation an arbitrarily small regularization parameter. The corresponding paraxial white-noise approximation that we derive is different from that of the previously studied high-frequency regime. Since the correlation length of the refractive index fluctuations due to atmospheric turbulence varies substantially, our results are relevant for numerous different propagation scenarios including microwave and radiowave propagation through various regions of the atmosphere. • Helmholtz equation in randomly layered media. • Stratified atmosphere, Rayleigh–Taylor turbulence and electron layers in ionosphere. • Electromagnetic propagation regimes cover microwave and radiowave frequencies. • Wavelengths are of the same order as the correlation length of random fluctuations. [ABSTRACT FROM AUTHOR]