On the justification of the Foldy-Lax approximation for the acoustic scattering by small rigid bodies of arbitrary shapes

We are concerned with the acoustic scattering problem by many small rigid obstacles of arbitrary shapes. We give a sufficient condition on the number $M$ and the diameter $a$ of the obstacles as well as the minimum distance $d$ between them under which the Foldy-Lax approximation is valid. Precisely, if we use single layer potentials for the representation of the scattered fields, as it is done sometimes in the literature, then this condition is $(M-1)\frac{a}{d^2} <c$, with an appropriate constant $c$, while if we use double layer potentials then a weaker condition of the form $\sqrt{M-1}\frac{a}{d} <c$ is enough. In addition, we derive the error in this approximation explicitly in terms of the parameters $M, a$ and $d$. The analysis is based, in particular, on the precise scalings of the boundary integral operators between the corresponding Sobolev spaces. As an application, we study the inverse scattering by the small obstacles in the presence of multiple scattering.Comment: arXiv admin note: substantial text overlap with arXiv:1308.307

Multiscale analysis of the acoustic scattering by many scatterers of impedance type

We are concerned with the acoustic scattering problem, at a frequency $\kappa$, by many small obstacles of arbitrary shapes with impedance boundary condition. These scatterers are assumed to be included in a bounded domain $\Omega$ in $\mathbb{R}^3$ which is embedded in an acoustic background characterized by an eventually locally varying index of refraction. The collection of the scatterers $D_m, \; m=1,...,M$ is modeled by four parameters: their number $M$, their maximum radius $a$, their minimum distance $d$ and the surface impedances $\lambda_m, \; m=1,...,M$. We consider the parameters $M, d$ and $\lambda_m$'s having the following scaling properties: $M:=M(a)=O(a^{-s})$, $d:=d(a)\approx a^t$ and $\lambda_m:=\lambda_m(a)=\lambda_{m,0}a^{-\beta}$, as $a \rightarrow 0$, with non negative constants $s, t$ and $\beta$ and complex numbers $\lambda_{m, 0}$'s with eventually negative imaginary parts. We derive the asymptotic expansion of the farfields with explicit error estimate in terms of $a$, as $a\rightarrow 0$. The dominant term is the Foldy-Lax field corresponding to the scattering by the point-like scatterers located at the centers $z_m$'s of the scatterers $D_m$'s with $\lambda_m \vert \partial D_m\vert$ as the related scattering coefficients.Comment: 27 pages, 1figur

Extraction of the index of refraction by embedding multiple and close small inclusions

We deal with the problem of reconstructing material coefficients from the farfields they generate. By embedding small (single) inclusions to these media, located at points $z$ in the support of these materials, and measuring the farfields generated by these deformations we can extract the values of the total field generated by these media at the points $z$. The second step is to extract the values of the material coefficients from these internal values of the total field. The main difficulty in using internal fields is the treatment of their possible zeros. In this work, we propose to deform the medium using multiple (precisely double) and close inclusions instead of only single ones. By doing so, we derive from the asymptotic expansions of the farfields the internal values of the Green function, in addition to the internal values of the total fields. This is possible because of the deformation of the medium with multiple and close inclusions which generates scattered fields due to the multiple scattering between these inclusions. Then, the values of the index of refraction can be extracted from the singularities of the Green function. Hence, we overcome the difficulties arising from the zeros of the internal fields. We test these arguments for the acoustic scattering by a refractive index in presence of inclusions modeled by the impedance type small obstacles.Comment: 21pages, 2 figures; Comparing to the previous version, we slightly modified the presentation and added a new section, section 3, discussing the stability issu

Location and size estimation of small rigid bodies using elastic far-fields

We are concerned with the linearized, isotropic and homogeneous elastic scattering problem by (possibly many) small rigid obstacles of arbitrary Lipschitz regular shapes in 3D. Based on the Foldy-Lax approximation, valid under a sufficient condition on the number of the obstacles, the size and the minimum distance between them, we show that any of the two body waves, namely the pressure waves P or the shear waves S, is enough for solving the inverse problem of detecting these scatterers and estimating their sizes. Further, it is also shown that the shear-horizontal part SH or the shear vertical part SV of the shear waves S are also enough for the location detection and the size estimation. Under some extra assumption on the scatterers, as the convexity assumption, we derive finer size estimates as the radius of the largest ball contained in each scatterer and the one of the smallest ball containing it. The two estimates measure, respectively, the thickness and length of each obstacle.Comment: 12pages. arXiv admin note: text overlap with arXiv:1308.307

The Foldy-Lax approximation of the scattered waves by many small bodies for the Lame system

We are concerned with the linearized, isotropic and homogeneous elastic scattering problem by many small rigid obstacles of arbitrary, Lipschitz regular, shapes in 3D case. We prove that there exists two constant $a_0$ and $c_0$, depending only on the Lipschitz character of the obstacles, such that under the conditions $a\leq a_0$ and $\sqrt{M-1}\frac{a}{d} \leq c_0$ on the number $M$ of the obstacles, their maximum diameter $a$ and the minimum distance between them $d$, the corresponding Foldy-Lax approximation of the farfields is valid. In addition, we provide the error of this approximation explicitly in terms of the three parameters $M, a$ and $d$. These approximations can be used, in particular, in the identification problems (i.e. inverse problems) and in the design problems (i.e. effective medium theory)

The equivalent medium for the elastic scattering by many small rigid bodies and applications

We deal with the elastic scattering by a large number $M$ of rigid bodies, $D_m:=\epsilon B_m+z_m$, of arbitrary shapes with $0<\textcolor{black}{\epsilon}<<1$ and with constant Lam\'e coefficients $\lambda$ and $\mu$. We show that, when these rigid bodies are distributed arbitrarily (not necessarily periodically) in a bounded region $\Omega$ of $\mathbb{R}^3$ where their number is $M:=M(\textcolor{black}{\epsilon}):=O(\textcolor{black}{\epsilon}^{-1})$ and the minimum distance between them is $d:=d(\textcolor{black}{\epsilon})\approx \textcolor{black}{\epsilon}^{t}$ with $t$ in some appropriate range, as $\textcolor{black}{\epsilon} \rightarrow 0$, the generated far-field patterns approximate the far-field patterns generated by an equivalent medium given by $\omega^2\rho I_3-(K+1)\mathbf{C}_0$ where $\rho$ is the density of the background medium (with $I_3$ as the unit matrix) and $(K+1)\mathbf{C}_0$ is the shifting (and possibly variable) coefficient. This shifting coefficient is described by the two coefficients $K$ and $\mathbf{C}_0$ (which have supports in $\overline{\Omega}$) modeling the local distribution of the small bodies and their geometries, respectively. In particular, if the distributed bodies have a uniform spherical shape then the equivalent medium is isotropic while for general shapes it might be anisotropic (i.e. $\mathbf{C}_0$ might be a matrix). In addition, if the background density $\rho$ is variable in $\Omega$ and $\rho =1$ in $\mathbb{R}^3\setminus{\overline{\Omega}}$, then if we remove from $\Omega$ appropriately distributed small bodies then the equivalent medium will be equal to $\omega^2 I_3$ in $\mathbb{R}^3$, i.e. the obstacle $\Omega$ characterized by $\rho$ is approximately cloaked at the given and fixed frequency $\omega$.Comment: 27pages, 2 figure

A cluster of many small holes with negative imaginary surface impedances may generate a negative refraction index

We deal with the scattering of an acoustic medium modeled by an index of refraction $n$ varying in a bounded region $\Omega$ of $\mathbb{R}^3$ and equal to unity outside $\Omega$. This region is perforated with an extremely large number of small holes $D_m$'s of maximum radius $a$, $a<<1$, modeled by surface impedance functions. Precisely, we are in the regime described by the number of holes of the order $M:=O(a^{\beta-2})$, the minimum distance between the holes is $d\sim a^t$ and the surface impedance functions of the form $\lambda_m \sim \lambda_{m,0} a^{-\beta}$ with $\beta >0$ and $\lambda_{m,0}$ being constants and eventually complex numbers. Under some natural conditions on the parameters $\beta, t$ and $\lambda_{m,0}$, we characterize the equivalent medium generating, approximately, the same scattered waves as the original perforated acoustic medium. We give an explicit error estimate between the scattered waves generated by the perforated medium and the equivalent one respectively, as $a \rightarrow 0$. As applications of these results, we discuss the following findings: 1. If we choose negative valued imaginary surface impedance functions, attached to each surface of the holes, then the equivalent medium behaves as a passive acoustic medium only if it is an acoustic metamaterial with index of refraction $\tilde{n}(x)=-n(x),\; x \in \Omega$ and $\tilde{n}(x)=1,\; x \in \mathbb{R}^3\setminus{\overline{\Omega}}$. This means that, with this process, we can switch the sign of the index of the refraction from positive to negative values. 2. We can choose the surface impedance functions attached to each surface of the holes so that the equivalent index of refraction $\tilde{n}$ is $\tilde{n}(x)=1,\; x \in \mathbb{R}^3$. This means that the region $\Omega$ modeled by the original index of refraction $n$ is approximately cloaked.Comment: arXiv admin note: text overlap with arXiv:1504.0694

The equivalent refraction index for the acoustic scattering by many small obstacles: with error estimates

Let $M$ be the number of bounded and Lipschitz regular obstacles $D_j, j:=1, ..., M$ having a maximum radius $a$, $a<<1$, located in a bounded domain $\Omega$ of $\mathbb{R}^3$. We are concerned with the acoustic scattering problem with a very large number of obstacles, as $M:=M(a):=O(a^{-1})$, $a\rightarrow 0$, when they are arbitrarily distributed in $\Omega$ with a minimum distance between them of the order $d:=d(a):=O(a^t)$ with $t$ in an appropriate range. We show that the acoustic farfields corresponding to the scattered waves by this collection of obstacles, taken to be soft obstacles, converge uniformly in terms of the incident as well the propagation directions, to the one corresponding to an acoustic refraction index as $a\rightarrow 0$. This refraction index is given as a product of two coefficients $C$ and $K$, where the first one is related to the geometry of the obstacles (precisely their capacitance) and the second one is related to the local distribution of these obstacles. In addition, we provide explicit error estimates, in terms of $a$, in the case when the obstacles are locally the same (i.e. have the same capacitance, or the coefficient $C$ is piecewise constant) in $\Omega$ and the coefficient $K$ is H\ddot{\mbox{o}}lder continuous. These approximations can be applied, in particular, to the theory of acoustic materials for the design of refraction indices by perforation using either the geometry of the holes, i.e. the coefficient $C$, or their local distribution in a given domain $\Omega$, i.e. the coefficient $K$.Comment: 22pages, 2 figure

The Foldy-Lax Approximation for the Full Electromagnetic Scattering by Small Conductive Bodies of Arbitrary Shapes

We deal with the electromagnetic waves propagation in the harmonic regime. We derive the Foldy-Lax approximation of the scattered fields generated by a cluster of small conductive inhomogeneities arbitrarily distributed in a bounded domain $\Omega$ of $\mathbb{R}^3$. This approximation is valid under a sufficient but general condition on the number of such inhomogeneities $m$, their maximum radii $\epsilon$ and the minimum distances between them $\delta$, the form $$(\ln m)^{\frac{1}{3}}\frac{\epsilon}{\delta} \leq C,$$ where $C$ is a constant depending only on the Lipschitz characters of the scaled inhomogeneities. In addition, we provide explicit error estimates of this approximation in terms of aforementioned parameters, $m, \epsilon, \delta$ but also the used frequencies $k$ under the Rayleigh regime. Both the far-fields and the near-fields (stated at a distance $\delta$ to the cluster) are estimated. In particular, for a moderate number of small inhomogeneities $m$, the derived expansions are valid in the mesoscale regime where $\delta \sim \epsilon$. At the mathematical analysis level and based on integral equation methods, we prove a priori estimates of the densities in the $L^{2,Div}_t$ spaces instead of the usual $L^2$ spaces (which are not enough). A key point in such a proof is a derivation of a particular Helmholtz type decomposition of the densities. Those estimates allow to obtain the needed qualitative as well as quantitative estimates while refining the approximation. Finally, to prove the invertibility of the Foldy-Lax linear algebraic system, we reduce the coercivity inequality to the one related to the scalar Helmholtz model. As this linear algebraic system comes from the boundary conditions, such a reduction is not straightforward

Characterization of the equivalent acoustic scattering for a cluster of an extremely large number of small holes

We deal with the time-harmonic acoustic waves scattered by a large number of small holes, of maximal radius $a, a<<1$, arbitrary (i.e. not necessarily periodically) distributed in a bounded part of a homogeneous background. We show that as their number $M$ grows following the law $M:=M(a):=O(a^{-s}), \; a<<1$, the collection of these holes has one of the following behaviors: 1. if $s<1$, then the scattered fields tend to vanish as $a$ tends to zero, i.e. the cluster is a soft one. 2. if $s=1$, then the cluster behaves as an equivalent medium modeled by a refraction index, supported in a given bounded domain $\Omega$, which is described by certain geometry properties of the holes and their local distribution. The cluster is a moderate (or intermediate) one. 3. if $s>1$, then the cluster behaves as a totally reflecting extended body, modeled by a bounded and smooth domain $\Omega$, i.e. the incident waves are totally reflected by the surface of this extended body. The cluster is a rigid one. These approximations are provided with explicit error estimates in terms of $a,\; a<<1$