Plane waves in a layered weakly dissipative randomly inhomogeneous medium

Abstract

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.     

Coherent backscattering of a light pulse by a slab of disordered medium

We obtain an exact analytical solution to the problem of the enhanced backscattering of a short pulsed signal from a two- and three-dimensional medium with isotropically scattering centres. The angular spectrum is expressed in terms of the solution to the corresponding stationary problem. The intensity oscillations are shown to appear on the tails of the angular spectrum. The origin of these oscillations is associated with the ballistic phase shift between the interfering waves arriving at the detector. It is shown that the finiteness of the slab thickness influences the magnitude of the backscattering intensity and does not change the shape of the angular spectrum. The range of validity for the diffusion approximation is pointed out.

The results obtained in the paper also contain a generalization of the well known solution to the problem of incoherent transfer to a pulsed signal to the case of two-dimensional disordered media.     

On the Milne Problem and the Hydrodynamic Limit for a Steady Boltzmann Equation Model

For a stationary nonlinear Boltzmann equation in a slab with a particular truncation in the collision operator, the Milne problem for the boundary layer together with a weak type of hydrodynamic behavior in the interior of the slab are studied by nonperturbative methods in the small-mean-free-path limit.     

Statistical modelling of the backscattered field in a one-dimensional non-stationary stochastic problem

Within the framework of an exact wave approach in the spatial-time domain, the one-dimensional stochastic problem of sound pulse scattering by a layered random medium is considered. On the basis of a unification of methods which has been developed by the authors, previously applied to the investigation of non-stationary deterministic wave problems and stochastic stationary wave problems, an analytical-numerical simulation of the behaviour of the backscattered field stochastic characteristics was carried out. Several forms of incident pulses and signals are analysed. We assume that random fluctuations of a medium are described by virtue of the Gaussian Markov process with an exponential correlation function. The most important parameters appearing in the problem are discussed; namely, the time scales of diffusion, pulse durations, the medium layer thickness or the largest observation time scale in comparison with the time scale of one correlation length for the case of a half-space. An exact pattern of the pulse backscattering processes is obtained. It is illustrated by the behaviour of the backscattered field statistical moments for all observation times which are of interest. It is shown that during the time interval when the main part of the pulse energy leaves the medium, the backscattered field is a substantially non-stationary process, having a non-zero mean value and an average intensity that decays according to a power law. There are various power indices for the different duration incident pulses, however, they are not the same as those of previous papers, which were obtained on the basis of an approximate and asymptotic analysis. We have also verified that the Gaussian law is valid for the probability density function of the backscattered field in the case of any incident pulse duration.     

低噪声通过特性的测量仿真研究

本文研究运用组合传感器和声压、振速联合信息处理技术进行低品声通过特性的测量。通过特性的背景噪声出非相干的各向同性白噪声和相干的交通干扰组成,交通干扰又分为平稳与非平稳两种,用平均声强器来抗非相干干扰;用交通干扰能量流中心方位抑制技术来抗平稳的相干干扰;对于非平稳的相干干扰而能量流中心方位抑制技术。仿真试验表明效果良好。     

Statistical modelling of the backscattered field in a one-dimensional non-stationary stochastic problem

Abstract

Within the framework of an exact wave approach in the spatial-time domain, the one-dimensional stochastic problem of sound pulse scattering by a layered random medium is considered. On the basis of a unification of methods which has been developed by the authors, previously applied to the investigation of non-stationary deterministic wave problems and stochastic stationary wave problems, an analytical-numerical simulation of the behaviour of the backscattered field stochastic characteristics was carried out. Several forms of incident pulses and signals are analysed. We assume that random fluctuations of a medium are described by virtue of the Gaussian Markov process with an exponential correlation function. The most important parameters appearing in the problem are discussed; namely, the time scales of diffusion, pulse durations, the medium layer thickness or the largest observation time scale in comparison with the time scale of one correlation length for the case of a half-space. An exact pattern of the pulse backscattering processes is obtained. It is illustrated by the behaviour of the backscattered field statistical moments for all observation times which are of interest. It is shown that during the time interval when the main part of the pulse energy leaves the medium, the backscattered field is a substantially non-stationary process, having a non-zero mean value and an average intensity that decays according to a power law. There are various power indices for the different duration incident pulses, however, they are not the same as those of previous papers, which were obtained on the basis of an approximate and asymptotic analysis. We have also verified that the Gaussian law is valid for the probability density function of the backscattered field in the case of any incident pulse duration.     

A restoration procedure for (non-stationary) signals from moving sources

Moving sources, such as trains, cars and airplanes, provide non-stationary sound and vibration signals in situations where the receiver is not moving with the source. For non-stationary signals there are strong limitations on the use of computerized analysis techniques based on Fourier transformation. For instance, it is not possible to compute reliably either power spectral density functions or coherence functions. A procedure has been developed, and is discussed in this paper, that restores non-stationary signals into stationary ones, thus enabling one to apply the analysis techniques mentioned above to moving source data with a reliable outcome.     

The plunging-mode response of an idealized airplane to atmospheric turbulent gusts

For the plunging-mode motion of an idealized airplane with pitch neglected, the response statistics have been formulated subject to atmospheric turbulence prescribed by the so-called Dryden's energy spectrum. The stationary input-response relation yields the following conclusions. (1) Atmospheric turbulent gusts should be modeled by incorporating not only the intensity modulation but also the time-varying energy spectrum. (2) The response exceedance curve can display convexity depending on the combination of gust intensity and spectrum parameter. (3) The integral length scale may be estimated alternately from the accelerometer data at the center of gravity and the gust intensity. As a consequence of conclusion (1), a discrete non-stationary gust model is proposed, consisting of a series of stationary processes, having different gust intensities and spectrum parameters. For such a modified Press' gust model, the response covariances of the plunging velocity and acceleration have been computed. Based on the elementary two-stationary-process gust model, sample calculations indeed produce convexity in the response exceedance curves, thereby substantiating (2). Finally, the implication of (3) is that one now has a means of checking consistency of the integral-length-scale measurements.     

Equivalent linear stochastic seismic response of isolated bridges

Stochastic response of bridges seismically isolated by lead-rubber bearings (LRB) is investigated. The earthquake excitation is modeled as a non-stationary random process (i.e. uniformly modulated broad-band excitation). The stochastic response of isolated bridge is obtained using the time-dependent equivalent linearization technique as the force-deformation behavior of the LRB is highly nonlinear. The non-stationary response of isolated bridge is compared with the corresponding stationary response in order to study the effects of non-stationary characteristics of the earthquake input motion. For a given isolated bridge system and excitation, it was observed that there exists an optimum value of the yield strength of LRB for which the root mean square (rms) absolute acceleration of bridge deck attains the minimum value. The optimum yield strength of LRB is investigated under important parametric variations such as isolation period and damping ratio of the LRB and the frequency content and intensity of earthquake excitation. It is shown that the above parameters have significant effects on the optimum yield strength of LRB. Finally, closed-form expressions for the optimum yield strength of LRB and corresponding response of the isolated bridge system are proposed. These expressions were derived based on the model of bridge with rigid deck and pier condition subjected to stationary white-noise excitation. It was observed that there is a very good comparison between the proposed closed-form expressions and actual optimum parameters and response of the isolated bridge system. These expressions can be used for initial optimal design of seismic isolation system for the bridges.     

Zur Theorie der Resonanzstreuung an Störstellen im Festkörper

On the Theory of Resonant Scattering at Impurity Centres in Solids For the case of a weak exciting field general formulae for describing both the stationary and the non-stationary behaviour of resonant scattering are derived. By taking into account that the dephasing of electronic transitions of impurity centres in solids cannot be described by constant phase relaxation rates the influence of one and two phonon processes on the stationary scattering spectrum is investigated.     

Collocation Methods for Hyperbolic Partial Differential Equations with Singular Sources

A numerical study is given on the spectral methods and the high order WENO finite difference scheme for the solution of linear and nonlinear hyperbolic partial differential equations with stationary and non-stationary singular sources. The singular source term is represented by the $δ$-function. For the approximation of the $δ$-function, the direct projection method is used that was proposed in [6]. The $δ$-function is constructed in a consistent way to the derivative operator. Nonlinear sine-Gordon equation with a stationary singular source was solved with the Chebyshev collocation method. The $δ$-function with the spectral method is highly oscillatory but yields good results with small number of collocation points. The results are compared with those computed by the second order finite difference method. In modeling general hyperbolic equations with a non-stationary singular source, however, the solution of the linear scalar wave equation with the non-stationary singular source using the direct projection method yields non-physical oscillations for both the spectral method and the WENO scheme. The numerical artifacts arising when the non-stationary singular source term is considered on the discrete grids are explained.     

Stationary Boltzmann equation for a degenerate gas in a slab: Boundary value problem and hydrodynamics

The boundary value problem for the stationary Boltzmann equation for a model gas in a plane slab is solved in full generality. The asymptotic behavior as the size of the slab goes to infinity is studied via a Chapman-Enskog expansion.     

A multi-dimensional differential approximation for absorbing/emitting and anisotropically scattering media with collimated irradiation

By considering the intensity within a medium to consist of a collimated and a fairly diffuse part, the overall problem of radiative transfer is reduced to two simpler ones: first the collimated intensity is obtained (equivalent in complexity to a nonscattering medium); for the evaluation of the diffuse part of the radiation (due to emission and scattering), a new differential approximation has been developed. To demonstrate the accuracy and simplicity of the present method, two sample cases are presented for which some exact solutions can be found in the literature: results are presented (i) for cosine-varying irradiation incident upon a two-dimensional, isotropically scattering slab and (ii) for irradiation with a Gaussian intensity distribution of a two-dimensional, anisotropically scattering semi-infinite cylindrical medium.     

On the theory of reflectivity and transmissivity of a lossless nonlinear dielectric slab

For the transverse electric polarization case (TE) we present a treatment of the optical reflectivity and transmissivity of a slab whose dielectric coefficient is a real valued function of the light intensity. If this function is numerically integrable with respect to the light intensity, our treatment can serve as an algorithm for a numerical solution of the nonlinear wave equation. If the dielectric function is proportional to the intensity, an analytical solution of the cubic wave equation is given for the electric field strength and for the phase of the field in terms of Weierstrass' elliptic functions and first elliptic theta functions, respectively. Evaluating this solution by means of a computer algebra system yields the reflectivity, transmissivity and phase dependency on the incident field intensity and on parameters characteristic for the problem. Certain combinations of the parameters lead to bistable and multivalued behavior. The solution found is used to determine the relative extrema of the reflectivity and the critical values of the thickness and of the incident intensity. The results are a generalization of linear optics results. Application of the analysis to the cubic-quintic wave equation yields the general analytic solution which is used to detemine the reflectivity of a semi-infinite nonlinear medium.     

非稳散斑场探测中的统计分析与评价

统计分析了非稳散斑场光强的空间场值系综均值与系统均值的联系，以此为依据评价了非稳散斑场的探测结果，并提出了改善探测置信度的方法。     

Variability of the vertical interference structure of the sound field in a shallow sea

Dynamics of the spectral intensity oscillations that occur in the vertical plane because of the time variability of the medium along the propagation path is described. The errors arising in measuring the frequency shifts of the interference structure are considered. For low-frequency broadband signals received on a stationary propagation path, experimental data on the shifts of their frequency spectra due to the variation of the reception depth are presented. The number of interfering modes and their arrival directions in the vertical plane are estimated from the measured frequency shifts of the spectral intensity oscillation.     

Mighty morphing spatial solitons and bullets

We give what we believe to be the first closed-form exact expression for the dynamic evolution of nonstationary beams of arbitrary intensity and width propagating in a uniform nonlinear medium and in both two and three dimensions. This shows that periodic and quasi-periodic (nonradiating) beams can exist in a non-Kerr nonlinear medium. The Schr?dinger equation is solved for Gaussian beams in a saturable medium. For one critical (initial) beam width, the Gaussian is a stable stationary soliton or bullet, independent of its intensity; otherwise, it breathes. New quasi-periodic beams (mighty morphing solitons) and bullets (mighty morphs) of elliptical cross section also exist whose ellipticity changes with propagation.     

宽频带光束非线性热像效应的实验研究

实验研究宽频带脉冲光束通过非线性介质时的非线性热像的形成过程,揭示了光束入射功率和脉冲宽度、介质厚度等因素对热像光强和位置的影响. 结果表明,与单纵模激光束一样,在宽频带脉冲光束情形下,非线性介质上游元件的模糊斑调制也可导致在下游共轭位置形成热像. 热像光强随入射光束功率的增加而增加;若脉冲能量一定,则热像光强随脉冲宽度的增加而减弱;若脉冲峰值功率一定,则热像光强随脉冲宽度的增加而增强. 关键词： 高功率激光 热像 宽带光束 小尺度自聚焦     

Half-range completeness for the Fokker-Planck equation

Half-range completeness theorems are proved for eigenfunctions associated to the one-dimensional Fokker-Planck equation in a semi-infinite medium. Existence and uniqueness results for perfectly absorbing, partially absorbing, and purely specularly reflecting boundary conditions are deduced for the stationary and time-dependent problems. Similar results are obtained for a slab geometry.