Time reversal super resolution in randomly layered media

In this paper we extend previous work on time reversal in randomly layered media [J.-P. Fouque, J. Garnier, A. Nachbin, K. Sølna, Time reversal refocusing for point source in randomly layered media, Wave Motion 42 (2005) 238–260]. We consider first the case of an active source embedded below the surface in a finely layered random medium. We carry out time reversal with a time reversal mirror placed at the surface and we consider here the case where this mirror is larger than the carrier wavelength. In contrast with the situation addressed in our previous paper, where the size of the mirror was comparable to the wavelength, we show that multi-pathing dramatically enhances the effective aperture of the mirror so that super resolution at the location of the source can be obtained. In other words, the focal spot radius of the refocused field obtained in the case of a multiply scattering medium is much smaller than the spot size obtained in the case of a homogeneous medium. This super resolution effect is obtained by time-reversing the long incoherent waves generated by the multiple scattering due to the thin layers. We also give an application to the problem of focusing on a passive scatterer buried in the random medium and illuminated by a source at the surface.     

Direct numerical modeling of time-reversal acoustic subwavelength focusing

Focusing waves back to their original source position is possible both experimentally and numerically thanks to time reversal mirrors (TRM). For a TRM placed in the far field of the source, the focusing spot of the reversed wavefield is subject to the diffraction limit and cannot be smaller than half the minimum wavelength, even for a very small source. Yet, numerous time reversal experiments in resonating media have shown subwavelength focusing. In this work, we show that it is possible to model these subwavelength focusing observations with simple physics, only the 2-D standard acoustic wave equation, and with specific fine scale heterogeneity. Our work is based on the spectral element method to solve the wave equation and to model time reversal experiments. Such a method makes it possible to propagate very long time series in complex and strongly discontinuous media with high accuracy. The acoustic wave equations are solved at the fine scale in media with one or more split rings of size much smaller than the wavelength. Such split rings produce a Helmholtz resonance effect as well as propagation band-gaps. We show that, in such media, even with a single split ring resonator, subwavelength focusing down to 1/13th of the minimum wavelength can be observed.     

A time-reversal method for an acoustical pulse propagating in randomly layered media

In the recent years a considerable amount of mathematical work has been devoted to the study of reflected signals obtained by the propagation of pulses in randomly layered media. We refer to [M. Asch, W. Kohler, G. Papanicolaou, M. Postel and B. White, “Frequency content of randomly scattered signals”, SIAM Review 33 (4), 519–625 (1991)] for an extensive survey and applications to inverse problems. The analysis is based on separation of scales between the correlation scale of the inhomogeneities present in the medium, the typical wavelengths of the pulse and the macroscopic variations of the medium. On the other hand, in the context of ultrasounds, time-reversal mirrors have been developed and their effects have been studied experimentally by Mathias Fink and his team at the Laboratoire Ondes et Acoustique (ESPCI-Paris). We refer to: [M. Fink, “Time reversal mirrors”, J. Phys. D: Appl. Phys. 26, 1333–1350 (1993)]. Our goal is to present a mathematical analysis of a time-reversal method for analyzing reflected signals in the model described in [M. Asch, W. Kohler, G. Papanicolaou, M. Postel and B. White, “Frequency content of randomly scattered signals”, SIAM Review 33(4), 519–625 (1991)]. We restrict our analysis to the one-dimensional case, the three-dimensional layered case being the content of a forthcoming paper. It is noticeable that we do not introduce new mathematics in the problem but simply put together an already existing mathematical theory and a new device, the time-reversal mirror.     

Statistical theory of radiative transfer in layered media

An equation for the probability density of the wave intensity which takes into account absorption, is obtained with a help of the invariant imbedding method. The limiting case when the medium occupies a half-space, is considered. The field intensity is found for the case of a source inside the medium. The conditions of applicability of the linear theory or radiative transfer are obtained. Numerical solutions of the equations corresponding to the statistical theory of radiative transfer in a layered medium with random inhomogeneities are discussed.     

Invariant imbedding method for wave problems

A new method, which reduces various boundary value problems for a wave equation to initial value problems, is developed. The scalar Helmholtz equation for the one- and three-dimensional cases is considered. The method is extended to the case of nonlinear media. Its applications to the wave equation for different dimensions and various media are described. The source of the wave field may be situated outside or inside the layer occupied by the medium. Governing equations are obtained for cases when one can neglect the backward scattering. The operators that arise are reduced to integral operators. The problem of wave scattering by a weakly rough surface is briefly considered.     

Wave dispersion in randomly layered materials

Wave scattering in materials composed of two kinds of alternating layers with different elastic properties and randomly distributed thicknesses has been modeled. The general form of the dispersion equation is derived for the unbounded layered medium. It defines two basic macroscopic characteristics of the scattered wave: phase velocity and attenuation, which are explicit functions of wave frequency and microscopic parameters of the system: acoustic properties of the layers and stochastic characteristics of their thickness distributions. The analytical expressions are derived for three special cases: for long waves; for a periodic medium composed of layers with constant thicknesses and for random medium with uniform distribution of layer thicknesses. Special attention is paid to the analysis of the frequency dependence of the wave parameters. It was shown that the predictions of the model for long waves and for periodic medium are compatible with the results obtained in the literature.Moreover, comparison of theoretical results for frequency dependent wave parameters with numerical simulations of pulse transmission through the slab of the randomly layered medium shows good qualitative and quantitative agreement in wide frequency range.     

Slip–Brinkman Flow Through Corrugated Microannulus with Stationary Random Roughness

This paper presents a boundary perturbation method of the Brinkman-extended Darcy model to investigate the flow in corrugated microannuli cylindrical tubes with slip surfaces. The stationary random model is used to mimic the surface roughness of the cylindrical walls. The tube is filled with a porous medium. We shall consider the two cases where corrugations are either perpendicular or parallel to the flow, and particular attention is given to the effect of the phase shift. The effects of the corrugations on the flow rate and pressure gradient are investigated as functions of wavelength, the permeability of the medium, the radius ratio and the slip parameter. Particular surface roughnesses are examined as special cases of stationary random surface. It is found that the effect of the partial slip is significant on the corrugation functions. The limiting cases of Stokes and Darcy’s flows and no-slip case are discussed.     

衰变热源作用下饱和多孔介质热固结问题的扩展精细积分法

热源作用下饱和多孔介质热固结效应是土木及能源工程领域的一个重要课题.由于问题的复杂性,已有的研究大多将介质假定为均匀各向同性,且将热源假定为恒定强度.实际工程中,天然饱和多孔介质常表现出明显的分层特性,热源强度也存在衰变性,为此本工作采用扩展精细积分法对衰变热源作用下层状饱和多孔介质的热固结问题进行研究.借助于积分变换,将饱和多孔介质热固结问题的偏微分方程转化为变换域内的常微分方程;然后对饱和多孔介质微层元进行合并消元,并结合边界条件,推导出衰变热源作用下层状饱和多孔介质热固结问题在积分变换域内的扩展精细积分解;对所得解答进行相应的数值积分逆变换,可获得所求温度、超静孔压及竖向位移在物理域内的解答.基于上述求解过程,编制相应的计算程序进行数值计算,通过与已有文献对比,验证本文扩展精细积分法在求解层状饱和多孔介质热固结问题中的适应性和正确性;最后通过几组算例,分析热源衰变周期、热源埋深及介质的成层性对热固结效应的影响.结果表明:热源衰变周期对温度和超静孔压的峰值、以及达到峰值的时间均有明显影响,衰变周期越长,二者峰值均越大,且达到峰值所需时间越长;热源埋深对超静孔压及竖向位移变化影响显著,深埋热源作用时热源两侧竖向位移呈对称分布,而浅埋热源两侧则无此现象;饱和多孔介质的分层特性对热固结效应影响明显.     

Three-dimensional steady-state closed form solution for multilayered fluid-saturated anisotropic finite media due to surface/internal point source

A three-dimensional(3 D)steady-state solution of fluid saturated anisotropic finite media is presented.The eigenequation method and the pseudo-Stroh formalism are used to obtain the exact solution for homogeneous saturated finite media.The propagator matrix method is introduced to deal with the corresponding multilayered poroelastic media.The poroelastic solutions due to surface or internal point fluid source are obtained.The comparison of the results of the saturated isotropic media in a half space and those obtained by the finite element method is given to illustrate the accuracy of the solution in a finite domain.Numerical solutions of a sandwich poroelastic medium are presented to analyze its hydromechanical behaviors.Two ratios of the horizontal permeability to vertical permeability and different source positions are investigated.The results show that the fluid parameters and source positions have great influence on the hydromechanical behaviors of the layered media.     

On the existence of diffusive waves in some non-linear unsteady flows of non-Newtonian fluids through porous media

We investigate the unsteady flow of power law fluids through porous media. We determine the pressure and velocity distributions when fluid is injected into a porous medium of infinite extend. We obtain solutions of progressive-wave type by means of a translation. We determine the necessary conditions for the existence of this type of solution regarding the prescribed pressure of injection and the initial pressure and velocity distributions in the porous medium. Similarity solutions are also obtained for the cases of a prescribed time dependent pressure of injection and a prescribed constant flow rate of injection. In the latter case the resulting ordinary differential equation is solved numerically. Point source solutions are also obtained for the case when an amount of fluid is instantaneously injected into the porous media. In all cases the rheological effects are presented and analyzed.