The Atkinson-Wilcox theorem in ellipsoidal geometry

The famous Atkinson-Wilcox theorem claims that any scattered field, no matter what the boundary conditions on the surface of the scatterer are, can be expanded into a uniformly and absolutely convergent series in inverse powers of distance and that once the leading coefficient of the expansion is known the full series can be recovered up to the smallest sphere containing the scatterer in its interior. The leading coefficient of the series is nothing else but the scattering amplitude. This is a very useful theorem, which provides the exact analogue of the Sommerfeld radiation condition, but it has the disadvantage of recovering the scattered field only outside the sphere circumscribing the scatterer. This means that an elongated obstacle which has a very large, as it compares to its volume, circumscribing sphere leaves a lot of exterior space where the scattered field cannot be recovered from its scattering amplitude. In the present work the Atkinson-Wilcox theorem has been extended to the ellipsoidal system where the theorem as well as the relative recovering algorithm holds true all the way down to the smallest circumscribing ellipsoid. Considering the anisotropic character of the ellipsoidal geometry it is obvious that an appropriately chosen ellipsoid can fit almost every smooth convex obstacle. Furthermore, such a result offers the best opportunity to develop a hybrid method based on the theory of infinite elements. Two orientations dependent differential operators are introduced in the recurrence scheme which, as the ellipsoid degenerates to a sphere, one of them vanishes, while the other reduces to the Beltrami operator. A reduction to spherical geometry is also included.     

Well‐conditioned boundary integral formulations for high‐frequency elastic scattering problems in three dimensions

We construct and analyze a family of well‐conditioned boundary integral equations for the Krylov iterative solution of three‐dimensional elastic scattering problems by a bounded rigid obstacle. We develop a new potential theory using a rewriting of the Somigliana integral representation formula. From these results, we generalize to linear elasticity the well‐known Brakhage–Werner and combined field integral equation formulations. We use a suitable approximation of the Dirichlet‐to‐Neumann map as a regularizing operator in the proposed boundary integral equations. The construction of the approximate Dirichlet‐to‐Neumann map is inspired by the on‐surface radiation conditions method. We prove that the associated integral equations are uniquely solvable and possess very interesting spectral properties. Promising analytical and numerical investigations, in terms of spherical harmonics, with the elastic sphere are provided. Copyright © 2014 John Wiley & Sons, Ltd.     

On the scattering frequencies of the laplace operator for exterior domains

In this paper we show that the so-called scattering frequencies of the Laplace operator over an exterior domain, subject to Robin or Dirichlet boundary condition, cannot lie in certain portions of the upper half-plane. The excluded sets depend only on the type of boundary condition and the radius of the smallest sphere containing the scattering obstacle.     

Low‐frequency scattering by a penetrable body with an eccentric soft or hard core

A plane wave is scattered by an acoustically soft or hard sphere, covered by a penetrable non‐concentric spherical lossless shell that disturbs the propagation of the incident wave field. The dimensions of the coated sphere are much smaller than the wavelength of the incident field. Low‐frequency theory reduces this scattering problem to a sequence of potential problems, which can be solved iteratively. Exactly one bispherical coordinate system exists that fits the given geometry of the obstacle. For the case of a soft and hard core, the exact low‐frequency coefficients of the zeroth and the first‐order for the near field as well as the first‐ and second‐order coefficients for the normalized scattering amplitude are obtained and the cross sections are calculated. Discussion of the results and their physical meaning is included. Copyright © 2009 John Wiley & Sons, Ltd.     

Non-linear, High Frequency Sound Waves

The theory of relatively undistorted waves (Varley & Cumberbatch,1966) is used to discuss finite amplitude, radially symmetric,isentropic waves in fluids. A simple asymptotic expansion whichgeneralizes that used in the linear theory of geometrical acousticsto take into account non-linear phenomena is given. The firstand second terms in this expansion are calculated. The firstterm agrees with a hypothesis of Whitham (1956). The theoryis used to discuss the flow produced by a pulsating sphere.     

空心球复合材料热弹性性质的一些精确结果

本文基于所提出的基体均匀场方法研究了空心球增强复合材料的热弹性性质·导出了均匀边界条件激发的局部热场和力学场量的关系，并进而得到了复合材料等效热弹性性质之间的精确关系·对于具有某种特定内外径比的空心球所构成的宏观各向同性复合材料，如果基体和空心球的热膨胀系数相同，可以证明其等效体积模量和线膨胀系可以精确地确定·     

Characterization of functions as radiation patterns in linear elasticity

In this paper a necessary and sufficient condition for a pair of vector functions to be radiation patterns is presented. More precisely, it is proved that two vector functions, the first in the radial direction and the second in the tangential one, are radiation patterns if and only if there are two entire harmonic vector functions whose radial and tangential projections, respectively, are identical with the previous functions on the unit sphere and whose L²-norm over a sphere of radius R is a function of exponential type in the variable R.     

Decay of Singular Values of Power Series Kernels on the Sphere

In this article, we provide decay rates for singular values of compact integral operators generated by power series kernels on either the unit sphere or the closed unit ball in ?^m + 1, m ≥ 1 under decay assumptions on the coefficients in the expansion of the kernel. The results are illustrated in concrete examples, including an integral operator generated by a Gaussian kernel on the sphere.     

Three dimensional electromagnetic scattering T-matrix computations

The infinite T-matrix method is a powerful tool for electromagnetic scattering simulations, particularly when one is interested in changes in orientation of the scatterer with respect to the incident wave or changes of configuration of multiple scatterers and random particles, because it avoids the need to solve the fully reconfigured systems. The truncated T-matrix (for each scatterer in an ensemble) is often computed using the null-field method. The main disadvantage of the null-field based T-matrix computation is its numerical instability for particles that deviate from a sphere. For large and/or highly non-spherical particles, the null-field method based truncated T-matrix computations can become slowly convergent or even divergent. In this work, we describe an electromagnetic scattering surface integral formulation for T-matrix computations that avoids the numerical instability. The new method is based on a recently developed high-order surface integral equation algorithm for far field computations using basis functions that are tangential on a chosen non-spherical obstacle. The main focus of this work is on the mathematical details required to apply the high-order algorithm to compute a truncated T-matrix that describes the scattering properties of a chosen perfect conductor in a homogeneous medium. We numerically demonstrate the stability and convergence of the T-matrix computations for various perfect conductors using plane wave incident radiation at several low to medium frequencies and simulation of the associated radar cross of the obstacles.     

On the smoothness of mean values of functions with summable spectral expansion

We consider spectral expansions associated with a self-adjoint extension of the Laplace operator in the n-dimensional domain. We show that if the spectral expansion of an arbitrary function at some point is summable by Riesz means, then its mean value over the sphere with center at that point has certain smoothness.     

Scattering of Plane Elastic Waves on a Small Obstacle Inside a Layer

The problem on the diffraction of a plane elastic wave with horizontal polarization (SH wave) on a small obstacle placed in a layer is investigated. The layer is situated on a half-space with stress-free boundary. The obstacle is assumed to be a circular cylinder with radius small in comparison with the length of the incident wave. The polarization of the incident wave is parallel to the axis of the cylinder. It is proved that the small inhomogeneity radiates as a linear source such that the intensity of the radiation is proportional to the area of the cross-section of the obstacle and the jump of the squared transverse velocities in the layer and in the obstacle. Bibliography: 5 titles.     

The Integral Form of the Equation of Radiative Transfer for an Inhomogeneous, Anisotropically Scattering, Solid Sphere

The integral form of the equation of radiative transfer is developedfor an absorbing, emitting, inhomogeneous, anisotropically scattering,solid sphere having internal energy sources, externally incidentradiation and a specularly or diffusively reflecting boundarysurface. The resulting integral form is useful for developingsolutions to radiation problems in a solid sphere by the applicationof projection techniques.     

Green’s function,temperature in a convectively cooled sphere with arbitrarily located spherical heat sources

Steady state heat conduction in a convectively cooled sphere having arbitrarily located spherical heat sources inside is treated with the method of Green’s function accompanied by a coordinate transform. Green’s function of the heat diffusion operator for a finite sphere with Robin boundary condition is obtained by spherical harmonics expansion. Verification of the analytical solution is exemplified in some generic cases related to the pebbles of South-African PBMR as of year 2000 with 268 MW thermal power. Analytical results for different sectors of the sphere (pebble) are compared with the results of computational fluid dynamics code FLUENT^™. This work is motivated through a modest effort to assess the stochastic effects of distribution and volumetric effects of fuel kernels within the pebbles of future-promising pebble bed reactors.     

On spherical-wave scattering by a spherical scatterer and related near-field inverse problems

A spherical acoustic wave is scattered by a bounded obstacle.A generalization of the ‘optical theorem’ (whichrelates the scattering cross-section to the far-field patternin the forward direction for an incident plane wave) is proved.For a spherical scatterer, low-frequency results are obtainedby approximating the known exact solution (separation of variables).In particular, a closed-form approximation for the scatteredwavefield at the source of the incident spherical wave is obtained.This leads to the explicit solution of some simple near-fieldinverse problems, where both the source and coincident receiverare located at several points in the vicinity of a small sphere.     

Reconstructing small perturbations of an obstacle for acoustic waves from boundary measurements on the perturbed shape itself

We derive relationships between the shape deformation of an impenetrable obstacle and boundary measurements of scattering fields on the perturbed shape itself. Our derivation is rigorous by using a systematic way, based on layer potential techniques and the field expansion (FE) method (formal derivation). We extend these techniques to derive asymptotic expansions of the Dirichlet-to-Neumann (DNO) and Neumann-to-Dirichlet (NDO) operators in terms of the small perturbations of the obstacle as well as relationships between the shape deformation of an obstacle and boundary measurements of DNO or NDO on the perturbed shape itself. All relationships lead us to very effective algorithms for determining lower order Fourier coefficients of the shape perturbation of the obstacle.     

Accuracy of the multipole expansion applied to a sphere in a creeping flow parallel to a wall

An example system is studied to discuss precision of the multipoleexpansion, applied to determine forces exerted on particlesby a viscous low-Reynolds-number fluid flow. A single spherein an ambient flow (pure shear, quadratic, and modulated shear)parallel to a close plane wall is considered. Forces and torquesexerted by the ambient flow on a motionless sphere are evaluated.Their precision is determined and related to a multipole orderof the truncation. Similar analysis is performed for a movingsphere with no ambient flow and for a freely moving sphere.Relative motion of the sphere with respect to the wall givesrise to strong lubrication interactions. It is analysed howthese interactions affect accuracy of the pure multipole expansion,and what are the smallest distances where it becomes insufficient.An alternative precise method is applied, in which lubricationexpressions are subtracted from the hydrodynamic forces andtorques, and the residue is evaluated as a fast-convergent seriesof inverse powers of the distance between the sphere centreand the wall. The accuracy of this procedure is carefully analysed.     

Drawdown in prolate spheroidal–spherical coordinates obtained via Green’s function and perturbation methods

When investigating aquifer behaviour it is important to note that there exists a close relationship between the geometrical properties of the aquifer and the behaviour of the solution. In this paper our concern is to solve the flow equation described by prolate spheroidal coordinates by means of perturbation and the Green’s function method, where the spheroid is considered to be a perturbation of a sphere. We transformed the spheroidal coordinates to spherical polar coordinates in the limit, as the shape factor tends to zero. The new groundwater flow equation is solved via an asymptotic parameter expansion and the Green’s function method. The approximate solution of the new equation is compared with experimental data from real world. To take into account the error committed while approximating, we estimate the error in the asymptotic expansion. The error functions obtained suggest that the error would be very small for the shape factor tending to zero if the first two terms of the expansion are taken as an approximation.