Many-body wave scattering problems in the case of small scatterers

Formulas are derived for solutions of many-body wave scattering problems by small particles in the case of acoustically soft, hard, and impedance particles embedded in an inhomogeneous medium. The case of transmission (interface) boundary conditions is also studied in detail. The limiting case is considered, when the size a of small particles tends to zero while their number tends to infinity at a suitable rate. Equations for the limiting effective (self-consistent) field in the medium are derived. The theory is based on a study of integral equations and asymptotics of their solutions as a→0. The case of wave scattering by many small particles embedded in an inhomogeneous medium is also studied.     

Electromagnetic scattering by a smooth convex impedance cone

The problem of the diffraction of an electromagnetic planewave by a convex cone of arbitrary smooth cross-section withimpedance (Leontovich) boundary conditions is studied. The vectorproblem is reduced to that for the Debye potentials. By meansof Kontorovich–Lebedev integrals, two spectral functionsare introduced and the corresponding boundary value problemis formulated. The spectral functions for the potentials arefound to satisfy the Helmholtz equations on the unit sphereand to be coupled through non-traditional boundary conditionsof the impedance type with shifts on the spectral variable.The use of the Green theorem permits us to establish an integralformulation of the boundary value problem for the spectral functions.The formal asymptotic solution of the problem is then givenfor the case of a narrow cone. For this, two different methodsare given: a method of perturbation applied to the spectralintegral equations and an adaptation of the method of matchingthe asymptotic series in spectral domain. Both methods leadto the same closed-form result for the leading term of the scatteringdiagram asymptotics.     

An analysis of the coupling of finite-element and Nystrm methods in acoustic scattering

In this paper we analyze the use of a combined Nystrm and finite-elementprocedure for approximating the scattered time-harmonic acousticfield produced when an incident wave interacts with a boundedinhomogeneity. The coupling technique uses a smooth artificialboundary and explicitly decouples the integral equation andfinite-element computations. We prove convergence of the method.One highlight of this analysis is that we prove Sobolev spaceerror estimates for the Nystrm scheme (which is usually analyzedin Hlder spaces). We also present some numerical results. E-mail: kirsch{at}am.uni-erlangen.de E-mail: monk{at}math.udel.edu     

Electromagnetic wave scattering by small impedance particles of an arbitrary shape

Scattering of electromagnetic (EM) waves by one and many small (ka?1) impedance particles D _m of an arbitrary shape, embedded in a homogeneous medium, is studied. Analytic formula for the field, scattered by one particle, is derived. The scattered field is of the order O(a ^2?κ ), where κ∈[0,1) is a number. This field is much larger than in the Rayleigh-type scattering. An equation is derived for the effective EM field scattered by many small impedance particles distributed in a bounded domain. Novel physical effects in this domain are described and discussed.     

Analytic properties of the scattering matrix of many particle systems

We generalize the previous results on a meromorphic continuation of the scattering matrix for N particle systems. Our proofs seem to be much simpler than those given before.Dedicated to the dear memory of david Milman     

反演声波阻尼系数的另一种方法

提出了一种方法,利用正则化方法和积分方程,由散射波的近场数据反演时间调和声波阻尼系数.给出了该方法收敛性的证明及数值例子,算法与数值例子表明这种方法不仅简单而且很有效.     

A numerical method for the analysis of the field scattering diagram on a system of impedance cylinders

Translated from Pryamye i Obratnye Zadachi Matematicheskoi Fiziki, pp. 168–180, 1991.     

Assessment of the surface radiation condition by reference to acoustic scattering by a hard sphere

Two forms of the surface radiation condition (SRC) introducedby different approaches are considered and their relative meritsare examined via a special problem, namely acoustic scatteringby a hard spherical object. Each of the methods offers a differentialequation on the scatterer to determine the surface field andeventually die calculation of the scattered field is reducedto quadra-tures. These equations are first solved exactly bya series expansion method for an incident plane wave and thefar field is obtained analytically. A comparison between theexact series solutions constructed by the SRC techniques andthe exact answer of the problem shows that both approaches,almost equivalently, provide the scattered field very accuratelyin the low and middle frequency ranges, i.e. for ka 5, but asthe frequency increases, although the phase remains remarkablyaccurate, the relative error in the far-field amplitude growsin the forward region; nevertheless, the results are still qualitativelyquite satisfactory and the accuracy increases with increasingfrequency in the backward direction. Since the series expansionmethod is limited to the geometries where the Helmholtz operatoris separable, some supplementary techniques are needed to applythe SRC concept for arbitrary convex objects. For this purpose,the effects of introducing a high-frequency asymptotic expansionand then an iterative technique for the solution of the SRCequations are investigated. The first-order approximations ofboth techniques also yield sufficiently accurate results forthe far field. Thus they appear to be reliable supplementarymethods for a general obstacle.     

An optimized finite element method for the analysis of 3D acoustic cavities with impedance boundary conditions

Numerical approaches studying the reduction of dispersion error for acoustic problems so far have focused on the models without impedance. Whereas, the practical acoustic problems usually involve impedance. This situation indicates that it is essential to study the numerical methods by taking into account the influence of impedance. In this work, an optimized finite element method is introduced to solve the three-dimensional steady-state acoustic problems with impedance. This technique resorts to heuristic optimization techniques to determine the integration points locations in elements. It develops a strategy to optimize the integration points locations, and makes use of adaptive genetic algorithm to achieve the best integration points locations for the construction of element matrix. By using the proposed method, a three-dimensional acoustic tube model with impedance is investigated, and the dispersion error, accuracy, convergence and efficiency of solutions are all compared to those of some existing numerical methods and reference solutions. Simultaneously, two practical cavity models are studied to verify the effectiveness and strongpoints of the proposed method as compared to existing numerical methods. Hence, the proposed method can be more widely applied to solve practical acoustic problems, yielding more accurate solutions.     

Passive impedance systems with losses of scattering channels

We consider a new model of passive impedance system with minimum losses of scattering channels and bilaterally stable evolutionary semigroup. In the case of discrete time, the passive linear stationary bilaterally stable impedance system Σ is regarded as a part of a certain minimal lossless transmission system with a -unitary system operator and a bilaterally (J ₁, J ₂)-inner (in a certain weak sense) transfer function in the unit disk whose 22-block coincides with the impedance matrix of the system Σ, belongs to the Carathéodory class, and possesses a pseudoextension. If the external space of the system Σ is infinite-dimensional, then, instead of the last property, we consider more complicated necessary and sufficient conditions for the impedance matrix of the system Σ. We study passive bilaterally stable impedance realizations with minimum losses of scattering channels (minimum, optimal, *-optimal, minimum and optimal, and minimum and *-optimal). __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 5, pp. 618–649, May, 2007.     

Reconstructing small perturbations of scatterers from electric or acoustic far‐field measurements

In this paper, we consider the problem of determining the boundary perturbations of an object from far‐field electric or acoustic measurements. Assuming that the unknown scatterer boundary is a small perturbation of a circle, we develop a linearized relationship between the far‐field data and the shape of the object. This relationship is used to find the Fourier coefficients of the perturbation of the shape. Copyright © 2008 John Wiley & Sons, Ltd.     

Detection of point-like scatterers using one type of scattered elastic waves

In this paper, we are concerned with the detection of point-like obstacles using elastic waves. We show that one type of waves, either the P or the S scattered waves, is enough for localizing the points. We also show how the use of S incident waves gives better resolution than the P waves. These affirmations are demonstrated by several numerical examples using a MUSIC type algorithm.     

用正则化方法求解声波散射反问题

研究了从声波散射场的远场模式的信息来再现散射物边界形状的反问题.首先构造表达散射物特征的指示函数,然后利用该函数之特性,建立求解该类反问题的基本方程,从而确定散射物的边界形状.在这个算法中,不需预先知道散射物的边界类型和形状等知识,从T ikhonov正则化方法进行的数值计算结果表明了该方法是有效的和实用的.     

Numerical solution of scattering problems using pseudo-differential impedance operators

Direct numerical solutions of scattering problems based on boundary-integral equations are computationally costly at high frequencies. A numerical method is presented that efficiently computes accurate approximations to unknown surface quantities given known surface data (an approximate Dirichlet-to-Neumann map). The method is based on a pseudo-differential impedance operator (PIO) numerically implemented using rational approximations. An example of a PIO is the so-called on-surface radiation condition (OSRC) method. For a convex obstacle, the method can be viewed as applying a parabolic equation directly on the surface of a scatterer. In contrast to past OSRCs, the use of rational approximations provides accuracy for wide scattering angles which is needed for grazing angles of incidence. Generalization to impedance operators for two-dimensional acoustic scatterers with concave parts is presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The direct electromagnetic scattering problem by a mixed impedance screen in chiral media

In this article solvability results for the direct electromagnetic scattering problem for a mixed perfectly conducting-impedance screen in a chiral environment is studied. In particular, incident time-harmonic electromagnetic waves in a chiral medium upon a partially coated open surface Γ (the ‘screen’), that satisfies an impedance boundary condition on one side and a perfectly conducting boundary condition on the other side, are considered. We introduce the Beltrami fields, appropriate boundary integral relations for these fields are proved and via them a uniqueness result is established. A variational method in a suitable functional space setting is considered and using a Calderon type operator for the chiral case, existence for the scattering problem is established.     

Surface wave scattering by a rectangular impedance cylinder located on a reactive plane

The electromagnetic scattering of the surface wave by a rectangular impedance cylinder located on an infinite reactive plane is considered for the case that the impedances of the horizontal and vertical sides of the cylinder can have different values. Firstly, the diffraction problem is reduced into a modified Wiener–Hopf equation of the third kind and then solved approximately. The solution contains branch‐cut integrals and two infinite sets of constants satisfying two infinite systems of linear algebraic equations. The approximate analytical or numerical evaluations of corresponding integrals and numerical solution of the linear algebraic equation systems are obtained for various values of parameters such as the surface reactance of the plane, the vertical and horizontal wall impedances, the width and the height of the cylinder. Copyright © 2004 John Wiley & Sons, Ltd.     

The factorization method for inverse acoustic scattering by a penetrable anisotropic obstacle

This paper is devoted to studying the factorization method applied to the inverse problem of reconstructing a penetrable anisotropic obstacle from far field patterns. We proved the validity of the factorization method. Copyright © 2013 John Wiley & Sons, Ltd.     

求解二维多区域声波散射问题的边界元方法

对于多散射区域的声波散射问题的外Neumann边值问题,用单层位势来逼近每个散射域上的散射波,再利用位势理论的跳跃关系将问题转换为第二类边界积分方程组的求解问题,然后用Nystrom方法进行了求解.对多个随机散射区域的声波散射问题,数值例子体现了该求解方法的可行性和准确性.     

The scattering of acoustic waves by a cylinder with a non-uniform elastic coating

The problem of the scattering of a plane acoustic wave by a solid cylinder with a radially non-uniform elastic coating is considered. An analytical expression describing the scattered acoustic field is obtained. The equations of motion of the non-uniform elastic cylindrical layer are reduced to a system of ordinary differential equations, the boundary-value problem for which is solved by the power-series method. The results of calculations of the directional pattern of the scattered field are presented.