A finite difference method for a coupled model of wave propagation in poroelastic materials

A computational method for time-domain multi-physics simulation of wave propagation in a poroelastic medium is presented. The medium is composed of an elastic matrix saturated with a Newtonian fluid, and the method operates on a digital representation of the medium where a distinct material phase and properties are specified at each volume cell. The dynamic response to an acoustic excitation is modeled mathematically with a coupled system of equations: elastic wave equation in the solid matrix and linearized Navier-Stokes equation in the fluid. Implementation of the solution is simplified by introducing a common numerical form for both solid and fluid cells and using a rotated-staggered-grid which allows stable solutions without explicitly handling the fluid-solid boundary conditions. A stability analysis is presented which can be used to select gridding and time step size as a function of material properties. The numerical results are shown to agree with the analytical solution for an idealized porous medium of periodically alternating solid and fluid layers.     

Evaluation of a wave-vector-frequency-domain method for nonlinear wave propagation

A wave-vector-frequency-domain method is presented to describe one-directional forward or backward acoustic wave propagation in a nonlinear homogeneous medium. Starting from a frequency-domain representation of the second-order nonlinear acoustic wave equation, an implicit solution for the nonlinear term is proposed by employing the Green's function. Its approximation, which is more suitable for numerical implementation, is used. An error study is carried out to test the efficiency of the model by comparing the results with the Fubini solution. It is shown that the error grows as the propagation distance and step-size increase. However, for the specific case tested, even at a step size as large as one wavelength, sufficient accuracy for plane-wave propagation is observed. A two-dimensional steered transducer problem is explored to verify the nonlinear acoustic field directional independence of the model. A three-dimensional single-element transducer problem is solved to verify the forward model by comparing it with an existing nonlinear wave propagation code. Finally, backward-projection behavior is examined. The sound field over a plane in an absorptive medium is backward projected to the source and compared with the initial field, where good agreement is observed.     

On wave solutions of a weakly nonlinear and weakly dispersive two-mode wave system

Abstract

In this paper, we introduce and study rigorously a Hamiltonian structure and soliton solutions for a weakly dissipative and weakly nonlinear medium that governs two Korteweg–de vries (KdV) wave modes. The bounded solution and traveling wave solution such as cnoidal wave and solitary wave are obtained. Subsequently, the equation is numerically solved by Fourier spectral method for its two-soliton solution. These solutions may be useful to explain the nonlinear dynamics of waves for an equation supporting multi-mode weakly dispersive and nonlinear wave medium. In addition, we give an explicit explanation of the mathematics behind the soliton phenomenon for a better understanding of the equation.     

Diffraction of electromagnetic plane wave by a slit in a homogeneous bi-isotropic medium

We investigated the diffraction of an electromagnetic plane wave by an infinite slit embedded in a homogeneous bi-isotropic medium. With the aim of deriving explicit expressions for the left- and right-handed Beltrami fields, we used the Fourier integral transform, the Wiener–Hopf technique and the steepest descent asymptotic method. The electric and magnetic fields, E and H, were determined from the Beltrami fields. Our graphical results indicate that the strength of both electric and magnetic fields reduces with the dissipation of bi-isotropic medium. While matching the diffraction pattern with the existing plane wave solution, the objective was, and is, to see how well spherical wave solution performs when it is developed for plane wave solution.     

Moment-screen method for wave propagation in a refractive medium with random scattering

Direct numerical solution of a parabolic equation (PE) for the second moment of the sound field in a refracting medium with random scattering is described. The method determines the mean-square sound pressure without requiring generation of random realizations of the propagation medium. The second-moment matrix is factored into components that are independently propagated with a conventional PE algorithm. A moment screen is periodically applied to attenuate the coherence of the wavefield, much as phase screens are often applied in the method of random realizations. An example involving upwind and downwind propagation in the near-ground atmosphere shows that the new direct method converges to an accurate solution faster than the method of random realizations and is particularly well suited to calculations at low frequencies.     

The problem of surface wave propagation in a reduced Cosserat medium

This article continues a series of publications devoted to the study of waves in the framework of the asymmetric theory of elasticity, where the deformed state of the medium is characterized by independent vectors of translation and rotation. The problem of acoustic Rayleigh wave propagation in half space is considered within a model of the reduced Cosserat medium. A general analytic solution of this problem is obtained. The analysis of this solution is compared with the corresponding solution for a classical elastic medium and full linear Cosserat medium. It is shown that the Rayleigh wave is characterized by a range of forbidden frequencies, where this wave cannot propagate. The dispersion curve consists of two branches. One of them has a cut-off frequency and cut-off wavenumber.     

A computational method for wave propagation from a point load in an anisotropic material

One approach which is employed to solve dynamic point load problems in plates and laminates is to take integral transforms to reduce the governing equations to a system of ordinary differential equations with respect to the depth variable. The solution of this system leads to expressions for the transforms of the displacement and stress components at any level in the plate and the transient response at any location may then be recovered by inversion of the multiple transforms. The formal transform inversion involves a double infinite integral but by making a change of variable this may be replaced by an infinite integral associated with a line source and a finite integral with respect to the orientation of the line. A first attempt at applying this approach to obtain the point load response of quasi-isotropic fibre composite laminate led to a non-causal predicted signal. This paper deals with an investigation of this proposed method applied to the simpler model problem of wave propagation in a two-dimensional anisotropic medium. Results are obtained for two different time histories of point loads, namely: a delta function; and a single period of a sine function. In the case of the delta function source a comparison is made with the analytic solution and the errors arising from the numerical approach are discussed. Graphs are also presented showing the non-causal contributions to the overall response which arise at individual angles of orientation of the line source.     

Multispeckle diffusing wave spectroscopy of colloidal particles suspended in a random packing of glass spheres

We use a multispeckle diffusing wave spectroscopy (MSDWS) method to study the ensemble-averaged dynamics of the fluctuating speckle pattern when illuminating colloidal particles suspended in a static and opaque porous medium with a coherent light source. Experiments were performed with Brownian latex particles in a random packing of glass spheres. The mixing of the light scattered by the moving colloidal particles and the porous matrix gives rise to a plateau value of the intensity autocorrelation function in the long-waiting-time limit. From the plateau in the correlation function, we can determine the fraction of light scattered from moving particles and estimate the photon mean free path in the colloidal solution. The method opens up promising possibilities to probe the static fraction in semisolid materials.     

Scattering and diffraction of a plane wave by a randomly rough half-plane

The scattering and diffraction of a TE (transverse electric) plane wave by a randomly rough half-plane are studied by a combination of three techniques: the Wiener-Hopf technique, the small perturbation method and a probabilistic method based on the shift-invariance of a homogeneous random function. By use of the D^a-Fourier transformation based on the shift-invariance, it is shown that the scattered wave is written by an inverse Fourier transformation of a homogeneous random function with a complex parameter. For a small rough case, such a random function with a complex parameter is expanded in a perturbation series and then the first-order solution is obtained exactly in an integral form. The first-order solution involves two physical processes such that the edge-diffracted wave is scattered by the randomly rough plane and the scattered wave, due to roughness, is diffracted by the half-plane. The solution is transformed into a sum of the Fresnel integrals with complex arguments, an integral along the steepest descent path and a branch-cut integral, which are evaluated numerically. Then, intensities of the coherently scattered wave and incoherent wave are calculated in the region near the edge and illustrated in figures.     

Scattering and diffraction of a plane wave by a randomly rough half-plane*

Abstract

The scattering and diffraction of a TE (transverse electric) plane wave by a randomly rough half-plane are studied by a combination of three techniques: the Wiener-Hopf technique, the small perturbation method and a probabilistic method based on the shift-invariance of a homogeneous random function. By use of the D^a-Fourier transformation based on the shift-invariance, it is shown that the scattered wave is written by an inverse Fourier transformation of a homogeneous random function with a complex parameter. For a small rough case, such a random function with a complex parameter is expanded in a perturbation series and then the first-order solution is obtained exactly in an integral form. The first-order solution involves two physical processes such that the edge-diffracted wave is scattered by the randomly rough plane and the scattered wave, due to roughness, is diffracted by the half-plane. The solution is transformed into a sum of the Fresnel integrals with complex arguments, an integral along the steepest descent path and a branch-cut integral, which are evaluated numerically. Then, intensities of the coherently scattered wave and incoherent wave are calculated in the region near the edge and illustrated in figures.     

Spectrum and wave shape of intense laser pulse propagating through Raman active medium

Results of the analytical solution of non-linear equations describing the propagation of an ultra-short intense laser pulse through a Raman-active medium are presented. The pulse passing through the medium generates a large number of high-order harmonics if the pulse intensity is close to some thresholds. The medium is described by the density matrix for two-level systems. The laser pulse duration was much shorter than both the period of oscillation and the relaxation times of an oscillator. Using this solution, the spectrum of such a pulse was expressed in terms of integrals of the initial field. The solution allows the harmonic spectrum dependence on the pulse intensity and effects of propagation to be investigated. Estimations of the conversion efficiency of the initial pumping frequency into high-order harmonics are given. The evolution of the pulse shape as a function of the distance passed by the pulse through the media was investigated. This revised version was published online in November 2006 with corrections to the Cover Date.     

Statistical characteristics of an intense wave behind a two-dimensional phase screen

An analysis of the parameters of nonlinear waves transmitted through a layer of a randomly inhomogeneous medium is carried out. The layer is modeled by a two-dimensional phase screen. Passing through the screen plane, the wave acquires a random phase shift. The wave front becomes distorted, and randomly located regions of ray convergence and divergence are formed, in which the nonlinear evolution of the wave alters profoundly. The problem is solved in the approximation of geometrical acoustics. The ray pattern of a plane wave transmitted through the regular screen is constructed. The solution that describes the spatial structure of the field and the evolution of an arbitrary temporal wave profile behind the screen is obtained. Statistical characteristics of the discontinuity amplitude are calculated for different distances from the screen. A random modulation is shown to result in a faster (in comparison with the case of a homogeneous medium) nonlinear attenuation of the wave and in the smoothing of the shock profile. The distribution function of the wave field parameters becomes broader because of random focusing effects.     

Fluctuations in the amplitude of a wave upon mutual diffusion of rays in a medium with random inhomogeneities

Fluctuations in the plane-wave amplitude and positions of two rays that propagate in a medium with random inhomogeneities of dielectric permittivity are considered. The solution to the problem is based on the ray-diffusion method. The Einstein-Fokker equation is obtained for the density of probabilities of the distance between the rays and relative amplitude in the case where the initial distance between the rays is much smaller than the correlation radius of dielectric permittivity. A conditional probability density for the relative amplitude is obtained provided that the distribution of distances between the rays is described by a logarithmic normal law, which does not take amplitude fluctuations into account. Numerical analysis of the analytical solution is carried out for the propagation of short radio waves in the troposphere and ionosphere.     

Focusing and dispersion of a plane wave by transversally isotropic elastic lenses

Transformation of a plane longitudinal wave front at the surfaces of a transversely isotropic elastic lens is considered. Nonlinear Snell equations are solved using an approach that combines Newton’s method and the algorithm of solution continuation with respect to a parameter. The cases of focusing and divergence of rays passing through a convex lens are investigated. Numerical examples are given for different relationships between the parameters of the elastic medium.     

Signature of wave localization in the time dependence of a reflected pulse

The average power spectrum of a pulse reflected by a disordered medium embedded in an N-mode waveguide decays in time with a power law t(-p). We show that the exponent p increases from 3 / 2 to 2 after N2 scattering times, due to the onset of localization. We compare two methods to arrive at this result. The first method involves the analytic continuation to an imaginary absorption rate of a static scattering problem. The second method involves the solution of a Fokker-Planck equation for the frequency dependent reflection matrix, by means of a mapping onto a problem in non-Hermitian quantum mechanics.     

Modal theory for the two-frequency mutual coherence function in random media: beam waves

Pulse propagation in a random medium is mainly determined by the two-frequency mutual coherence function which satisfies the parabolic equation. It has recently been shown that this equation can be solved by separation of variables, thereby reducing the solution for any structure function to the solution of ordinary differential equations. In this paper, the method is applied for a beam-wave excitation in a random medium. The exact solution for a quadratic medium is derived. For non-quadratic power-law media an analytical expression at equal positions is presented.     

Elastic wave propagation in a randomly stratified solid medium

The mean-field method is used to analyse longitudinal and transverse (both SV- and SH-type) wave propagation in an unbounded randomly stratified solid medium. It is assumed that elastic moduli of the medium are constant while a density is a random function of the cartesian coordinate z. For a case of small density fluctuations, expressions are obtained for z-components of effective propagation vectors of P-, SV- and SH-waves for arbitrary relations between wavelengths and a correlation length of the random inhomogeneities. It is shown, that when the correlation length is small in comparison with the wavelengths, the mean-field attenuation coefficients are proportional to the frequency squared. In this case P- and SV-waves convert into each other. When the correlation length is large in comparison with the wavelengths, the mean-field attenuation coefficients are also proportional to the frequency squared, but in this case P- and SV-waves propagate independently.