Approximation of the parabolic equation and the wavefield of a point source in a layered random medium

Abstract

This paper studies the wavefield of a source in a multidimensional randomly layered medium. They obtained asymptotical expressions of the wave statistical characteristics for different boundary conditions both in the framework of the parabolic equation approximation and the exact formulation of the boundary problem for the Helmholtz equation. It is shown that the presence of a small but finite absorption γ is most important for the statistics. The diffraction effects turn out to be like those of absorption, but γ cannot tend to zero in this problem. In an appendix they give the factorization formulae of the wave equation solution in a layered medium.     

Influence of boundary conditions on statistical characteristics of the wavefield in a layered randomly inhomogeneous medium

Abstract

We consider the problem of the oblique incidence of a wave on the half-space of a layered randomly inhomogeneous medium without absorption and analyse the influence of boundary conditions on the probability distribution of the reflection coefficient phase, and the statistical behaviour of the wavefield intensity moments inside the medium.     

Influence of boundary conditions on statistical characteristics of the wavefield in a layered randomly inhomogeneous medium

We consider the problem of the oblique incidence of a wave on the half-space of a layered randomly inhomogeneous medium without absorption and analyse the influence of boundary conditions on the probability distribution of the reflection coefficient phase, and the statistical behaviour of the wavefield intensity moments inside the medium.     

Exact solution of a massless scalar field with a relevant boundary interaction

We solve exactly the “boundary sine-Gordon” system of a massless scalar field with a potential at a boundary. This model has appeared in several contexts, including tunneling between quantum-Hall edge states and in dissipative quantum mechanics. For β² < 8π, this system exhibits a boundary renormalization-group flow from Neumann to Dirichlet boundary conditions. By taking the massless limit of the sine-Gordon model with boundary potential, we find the exact S-matrix for particles scattering off the boundary. Using the thermodynamic Bethe ansatz, we calculate the boundary entropy along the entire flow. We show how these particles correspond to wave packets in the classical Klein-Gordon equation, thus giving a more precise explanation of scattering in a massless theory.     

The exact response of a two-dimensional flat-layered medium to a probing pulse: An application to a layer stripping reconstruction procedure

Abstract

We consider a simple model problem that can be found in many fields of application such as, for example, reflection seismology. That is we consider an initial boundary value problem on a half-plane for a class of two-dimensional wave equations with a piecewise-constant coefficient. This coefficient describes the flat layered medium under consideration. An initial pulse located on the boundary of the half-plane is used to probe the medium. An integral representation of the solution of this problem is obtained by studying the spectral measures of some differential operators in one variable. This integral representation is exploited to obtain an ‘explicit’ formula for the solution of the problem considered evaluated at the location of the probing pulse. This ‘explicit’ formula is exploited to reconstruct the structure of the medium from its response to a probing pulse via a layer stripping procedure. Some numerical results obtained with this procedure on test problems are shown. The ‘explicit’ formula obtained can be used in several other contexts such as, for example, the study of perturbed flat layered media or the study of random flat layered media.     

Effect of pulse entrapping on diffuse reflection from a resonant random medium: exact solution to the scalar albedo problem

A recently derived radiative transfer equation with three Lorentzian delay kernels is applied to an albedo problem of the scalar wave field produced by the diffuse reflection of a quasi-monochromatic pulse from a semi-infinite random medium consisting of resonant point-like scatterers. The albedo problem is solved exactly in terms of the Chandrasekhar H-function H(μλ), extended analytically into the complex single-scattering albedo λ plane. The resulting analytic solution for the time evolution of a diffusely reflected short pulse is used to study on the whole time axis the effect of the redistribution of the energy of the propagated pulse from the front to the rear of the pulse in cases where the pulse may for most of the propagation time through the medium be 'entrapped' inside resonant scatterers. By considering the power flux through unit area of the boundary of the medium and unit solid angle, it is shown that the relative shift of an 'energy centroid' ('centre of mass') of the pulse diffusely reflected from the resonant random medium (compared with the pulse energy centriod in the non-resonant case) is equal to the parameter describing the energy accumulation inside the scatterers. This result may be used for experimental study of resonant random media with the aid of a short-pulse technique.     

η in the nuclear medium within a chiral unitary approach

The s-wave η self-energy in the nuclear medium is calculated in a chiral unitary approach. A coupled channel Bethe–Salpeter equation is solved to obtain the effective ηN interaction in the medium. The base model reproduces well the free space πN elastic and inelastic scattering at the ηN threshold or N^*(1535) region. The Pauli blocking on the nucleons, binding potentials for the baryons and self-energies of the mesons are incorporated, including the η self-energy in a self-consistent way. Our calculation predicts about −54−i29 MeV for the optical potential at normal nuclear matter for an η at threshold but also shows a strong energy dependence of the potential.     

Instability of oscillations in the flow moving through a random medium as the counterpart of the localization of waves in a passive random medium

This paper deals with waves propagating in a one-dimensional flow moving through a randomly layered medium. The flow velocity is assumed to be greater than the group velocity of the waves in the reference system of the flow. As a result, in the laboratory reference system, all the waves propagate in a single direction. Amplitudes of these waves moving through a randomly inhomogeneous medium are growing exponentially. This effect is analogous to the wave localization phenomenon in a randomly inhomogeneous passive medium. The only difference is that the wave propagation in a passive medium is described by the boundary value problem, while all the oscillations in a medium with flow propagate in a single direction and hence the corresponding problem is formulated in the form of the initial value Cauchy problem. In the former case exponentially decreasing solutions are realized in the direction of the wave incidence, while in the latter case (as in the case of parametric resonances) the exponentially increasing solutions are realized.     

On the problem of analyzing the dynamic properties of a layered half-space

An efficient method is developed for constructing the Green matrix functions of a layered inhomogeneous half-space. Matrix formulas convenient for programming are proposed, which make it possible to study the properties of a multilayered half-space with high accuracy. As an example of the problem of oscillations of a three-layer half-space, transformation of the dispersion characteristics of a three-layered medium is shown as a function of the relations of the mechanical and geometric parameters of its components. Study of the properties of the Green’s function of a medium with a low-velocity layered inclusion showed that each mode of a surface wave exists in a limited frequency range: in addition to the critical frequency of mode occurrence, the frequency of its disappearance exists—a frequency above which the mode is suppressed because of superposition of the zero of the Green’s function on its pole. A similar study conducted for a medium with a high-velocity layered inclusion has shown that in addition to the cutoff frequency (the frequency at which a surface wave propagating in the low-frequency range disappears), there is the frequency of its recurrent generation—the upper boundary of the “cutoff range” of the first mode. Beyond this range, the first mode propagates, and also the other propagating modes can appear. The critical relation of the geometric parameters of the medium determining the existence and boundaries of the cutoff range of a wave is established.     

一阶速度-应力Biot双相各向同性介质弹性波

提出一种等价的一阶双曲型速度一应力Biot双相各向同性介质弹性波波动方程,以实现双相介质混合波场中纯快慢纵波和纯横波波场分离的问题.应用散度和旋度理论证明双相介质等价方程波场分离的可行性,采用高阶交错网格有限差分法构建高精度正演算子,推导其PML吸收边界条件和稳定性条件,并对均匀双相介质和层状非均匀双相介质模型进行数值...     

Sound-absorbing media with two types of acoustic losses

Sound-absorbing media with two types of acoustic losses, namely, inertial and deformation are considered. Such a medium is described by a complex density and a complex compressibility. It is shown that a certain relation between the latter quantities imparts some specific properties to the medium: a real wave impedance, complete absorption of sound waves at the boundary with the external medium, the properties of a viscous or “superviscous” medium, etc. The results are generalized to the case of a layered inhomogeneous medium with two types of losses.     

Instability of Optical Solitons in the Boundary Value Problem for a Medium of Finite Extension

We consider an integrable nonlinear wave system (anisotropic chiral field model) which exhibits a soliton solution when the Cauchy problem for an infinitely long medium is posed. Whenever the boundary value problem is formulated for the same system but for a medium of finite extension, we reveal that the soliton becomes unstable and the true attractor is a different structure which is called polarization attractor. In contrast to the localized nature of solitons, the polarization attractor occupies the entire length of the medium. By demonstrating the qualitative difference between nonlinear wave propagation in an infinite medium and in a medium of finite extension (with simultaneous change of the initial value problem to the boundary value problem), we would like to point out that solitons may loose their property of being stable attractors. Additionally, our findings show the interest of developing methods of integration for boundary value problems.     

Plane waves in a layered weakly dissipative randomly inhomogeneous medium

In the framework of the embedding method the authors consider the stationary and non-stationary problem of a plane-wave incident on a randomly inhomogeneous medium. For the stationary problem there are three regions of sufficiently different behaviour of the wavefield intensity moments inside a weakly dissipative medium. For the non-stationary problem they succeeded in calculating the average intensity at t→+∞ by means of analytical prolongation of the stationary problem solution with respect to the absorption parameter. The time asymptotic of the averaged intensity on the boundary slab is also obtained for a finite-thickness slab.     

双折射吸收非线性介质薄膜中倍频的产生

从二次谐波电场满足的波动方程出发，考虑到介质对二次谐波的吸收以及把完全边界条件应用于基频波和二次谐波在介质的入射面和出射面之间的反射效应，推导出了单光轴非线性薄膜介质中的二次谐波输出功率的计算式.结果表明：当采用复折射率的概念时，可以把非线性无吸收介质的二次谐波输出功率计算式应用到非线性吸收介质二次谐波输出功率计算式中去.     

双折射吸收非线性介质薄膜中倍频的产生

从二次谐波电场满足的波动方程出发,考虑到介质对二次谐波的吸收以及把完全边界条件应用于基频波和二次谐波在介质的入射面和出射面之间的反射效应,推导出了单光轴非线性薄膜介质中的二次谐波输出功率的计算式.结果表明：当采用复折射率的概念时,可以把非线性无吸收介质的二次谐波输出功率计算式应用到非线性吸收介质二次谐波输出功率计算式中去. 关键词： 非线性 倍频 二次谐波功率     

Normal skin effect in the polycrystals

The surface impedance γ and the penetration depth δ of the electromagnetic field was calculated for a polycrystalline conductor under the conditions of the normal skin effect. The polycrystal is composed from the single-phase slightly anisotropic single-crystalline grains of arbitrary symmetry. Corrections (γ-γ₀)/γ₀ and (δ-δ₀)/δ₀ due to the polycrystallinity are functions of the parameter ζ=δ₀/a, where γ₀ and δ₀ are the impedance and the penetration depth of the isotropic conductor with the conductivity σ₀=Sp σ_ik/3, σ_ik is the separate grain conductivity tensor, and a is the mean size of a grain.

The effective conductivity of a polycrystal can be obtained either from the impedance γ (σ_ef^(γ)) or from the penetration depth δ (σ_ef^(δ)) It was found that σ_ef^(γ)≠σ_ef^(δ) The corrections (σ-σ_ef^((γ))/σ₀ and (σ₀-σ_ef^(δ))/σ₀ depend on the frequency of the incident wave.

The obtained results are also applicable in the infrared spectrum region. It was shown that, just as for the isotropic conductor, there is the total reflection of the electromagnetic wave from the polycrystal surface.

It was necessary to examine the oblique incidence of an electromagnetic wave onto the surface of an isotropic metal for the substantiation of the set of the boundary conditions used.     

An Inverse Problem of Determining Coefficients in a One-Dimensional Radiative Transport Equation

We consider an inverse problem of determining unknown coefficients for a one-dimensional analogue of radiative transport equation. We show that some combination of the unknown coefficients can be uniquely determined by giving pulse-like inputs at the boundary and observing the corresponding outputs. Our result can be applied for determination of absorption and scattering properties of an optically turbid medium if the radiative transport equation is appropriate for describing the propagation of light in the medium.     

Phase fluctuations and localization length in layered randomly inhomogeneous media

We consider the problem of the inclined incidence of a plane wave on the halfspace of a randomly inhomogeneous layered medium without absorption. The probability distribution of the reflection coefficient phase with respect to the incidence angle is calculated. With the help of this distribution we analyse the quantities connected with localisation effects—the length localization and moments of the intensity inside the medium.     

Acoustic wave propagation simulation in a poroelastic medium saturated by two immiscible fluids using a staggered finite-difference with a time partition method

Based on the three-phase theory proposed by Santos, acoustic wave propagation in a poroelastic medium saturated by two immiscible fluids was simulated using a staggered high-order finite-difference algorithm with a time partition method, which is firstly applied to such a three-phase medium. The partition method was used to solve the stiffness problem of the differential equations in the three-phase theory. Considering the effects of capillary pressure, reference pressure and coupling drag of two fluids in pores, three compressional waves and one shear wave predicted by Santos have been correctly simulated. Influences of the parameters, porosity, permeability and gas saturation on the velocities and amplitude of three compressional waves were discussed in detail. Also, a perfectly matched layer (PML) absorbing boundary condition was firstly implemented in the three-phase equations with a staggered-grid high-order finite-difference. Comparisons between the proposed PML method and a commonly used damping method were made to validate the efficiency of the proposed boundary absorption scheme. It was shown that the PML works more efficiently than the damping method in this complex medium. Additionally, the three-phase theory is reduced to the Biot’s theory when there is only one fluid left in the pores, which is shown in Appendix. This reduction makes clear that three-phase equation systems are identical to the typical Biot’s equations if the fluid saturation for either of the two fluids in the pores approaches to zero. Supported by the Key Program of the National Natural Science Foundation of China (Grant No. 10534040) and the National Natural Science Foundation of China (Grant No. 10674148)     

Phase fluctuations and localization length in layered randomly inhomogeneous media

Abstract

We consider the problem of the inclined incidence of a plane wave on the halfspace of a randomly inhomogeneous layered medium without absorption. The probability distribution of the reflection coefficient phase with respect to the incidence angle is calculated. With the help of this distribution we analyse the quantities connected with localisation effects—the length localization and moments of the intensity inside the medium.