Scattering of a spheroidal particle illuminated by a gaussian beam

An approach to expanding a Gaussian beam in terms of the spheroidal wave functions in spheroidal coordinates is presented. The beam-shape coefficients of the Gaussian beam in spheroidal coordinates can be computed conveniently by use of the known expression for beam-shape coefficients, g(n), in spherical coordinates. The unknown expansion coefficients of scattered and internal electromagnetic fields are determined by a system of equations derived from the boundary conditions for continuity of the tangential components of the electric and magnetic vectors across the surface of the spheroid. A solution to the problem of scattering of a Gaussian beam by a homogeneous prolate (or oblate) spheroidal particle is obtained. The numerical values of the expansion coefficients and the scattered intensity distribution for incidence of an on-axis Gaussian beam are given.     

The analytic solution for hydrodynamic diffraction by submerged prolate spheroids in infinite water depth

This study investigates the linear hydrodynamic scattering problem by stationary prolate spheroidal bodies and aims at providing an analytic solution for the associated boundary value problem. It extends the work of the present author on the hydrodynamics of oblate spheroidal bodies following the same procedure. The structural model under consideration is a spheroid with its polar axis greater than its equatorial diameter, subjected to the action of monochromatic incident waves. The polar axis is assumed to be perpendicular to the free surface that leads to the axisymmetric case concept. The analytic solution is sought using the method of multipole expansions constructed by employing Thorne’s formulas (Multipole expansions in the theory of surface waves. Proc Cam Philos Soc 49:707–716, 1953) that describe the velocity potential at singular points within a fluid domain with free upper surface and infinite water depth. The final stage of the solution process is the application of the zero velocity condition on the wetted surface of the spheroid. Inevitably this task requires the transformation of the involved velocity potentials, originally expressed with respect to spherical and polar coordinates, into prolate spheroidal coordinates. To this end, the appropriate addition theorems are derived, which recast Thorne’s expressions into infinite series of associated Legendre functions.     

Scattering of a point generated field by a multilayered spheroid

Summary The point source excitation acoustic scattering problem by a multilayer isotropic and homogeneous spheroidal body is presented. The multilayer spheroidal body is reached by an acoustic wave emanated by an external point source. The core spheroidal region is inpenetrable and rigid. The exterior interface and the interfaces separating the interior layers are penetrable. The scattered field is determined given the geometrical and physical characteristics of the spheroidal body, the location of the point source and the form of the incident field. The approach is not limited in a certain region of frequencies.     

Electromagnetic fields for a spheroidal particle with an arbitrary embedded source

A spheroidal coordinate separation-of-variables solution has been developed for the determination of the internal, near-surface, and scattered fields of a spheroid (either prolate or oblate) with an embedded source of arbitrary type, location, and orientation. Presented results for (1,0) and (1,1) electric multipoles embedded in 2:1 axis ratio prolate and oblate spheroids (equal volume sphere size parameter equal to 20) illustrate that the presence of the spheroid interface can have a profound effect on the corresponding far-field scattering pattern. The calculation procedure could be used, for example, to model the emission of inelastic scattered light (Raman, fluorescence, etc.) from biological particles of appreciably elongated (prolatelike) or appreciably flattened (oblatelike) geometries.     

Light scattering by a multilayered spheroidal particle

The light scattering problem for a confocal multilayered spheroid has been solved by the extended boundary condition method with a corresponding spheroidal basis. The solution preserves the advantages of the approach applied previously to homogeneous and core-mantle spheroids, i.e., the separation of the radiation fields into two parts and a special choice of scalar potentials for each of the parts. The method is known to be useful in a wide range of the particle parameters. It is particularly efficient for strongly prolate and oblate spheroids. Numerical tests are described. Illustrative calculations have shown that the extinction factors converge to average values with a growing number of layers and how the extinction varies with a growth of particle porosity.     

Generalized Lorenz-Mie theory for an arbitrarily oriented, located, and shaped beam scattered by a homogeneous spheroid

The theory of an arbitrarily oriented, shaped, and located beam scattered by a homogeneous spheroid is developed within the framework of the generalized Lorenz-Mie theory (GLMT). The incident beam is expanded in terms of the spheroidal vector wave functions and described by a set of beam shape coefficients (G(m)(n),(TM),G(m)(n),(TE)). Analytical expressions of the far-field scattering and extinction cross sections are derived. As two special cases, plane wave scattering by a spheroid and shaped beam scattered by a sphere can be recovered from the present theory, which is verified both theoretically and numerically. Calculations of the far-field scattering and cross sections are performed to study the shaped beam scattered by a spheroid, which can be prolate or oblate, transparent or absorbing.     

Scattering of a spherical acoustic field from an eccentric spheroidal structure simulating the kidney-stone system

Summary.  We consider the acoustic scattering of time-harmonic spherical waves from an eccentric non coaxial spheroidal structure simulating the kidney-stone system. The proposed analysis is based on the application of the translational addition theorem for spheroidal wave functions. The resulting theoretical model is frequency-independent. Numerical results concerning the applicability of our approach are also presented. Received September 20, 2002 Published online: March 20, 2003     

Scattering of electromagnetic waves by a perfect electromagnetic conducting sphero

An exact solution to the problem of scattering of electromagnetic waves from a perfect electromagnetic conducting spheroid is presented, using the method of separation of variables. The formulation of the problem is realised by expanding the incident as well as the scattered electromagnetic fields in terms of appropriate spheroidal vector wave functions and imposing the appropriate boundary conditions at the surface of the spheroid. This generates a set of simultaneous equations, the solution of which yields the unknown coefficients associated with the expansion of the scattered electromagnetic field. Results are presented in the form of normalised bistatic and backscattering cross-sections for spheroids of different axial ratios, sizes and admittances, for both transverse electric and transverse magnetic polarisations of the incident wave.     

Narrow resonances and ripple fluctuations in light scattering by a spheroid

We develop a semiclassical theory to explain the rapid ripple fluctuations in the extinction efficiency of light scattering by a transparent prolate spheroid. The theory is based on uniform asymptotic expansion of spheroidal radial functions. We have calculated the extinction efficiency for normal and oblique incidence. Our results suggest that the excitation of resonant electromagnetic modes inside a spheroidal particle is an important factor in the ripple structure. To verify this assumption and based on a Breit-Wigner formula, we develop a method to fit the peaks that appear in the spheroid's extinction cross section when some scattering parameters vary. In other words, our calculations suggest that narrow resonances are related to ripple fluctuations, whereas broad resonances contribute to extinction cross-sectional background.     

On connections between boundary integral equations and T-matrix methods

Acoustic scattering by three-dimensional obstacles is considered, using boundary integral equations, null-field equations and the T-matrix. Connections between these techniques are explored. It is shown that solving a boundary integral equation by a particular Petrov–Galerkin method leads to the same algebraic system as obtained from the null-field equations. It is also emphasised that the T-matrix can be constructed by solving boundary integral equations rather than by solving the null-field equations.     

Degenerate scale for the analysis of circular thin plate using the boundary integral equation method and boundary element methods

In this paper, the degenerate scale for plate problem is studied. For the continuous model, we use the null-field integral equation, Fourier series and the series expansion in terms of degenerate kernel for fundamental solutions to examine the solvability of BIEM for circular thin plates. Any two of the four boundary integral equations in the plate formulation may be chosen. For the discrete model, the circulant is employed to determine the rank deficiency of the influence matrix. Both approaches, continuous and discrete models, lead to the same result of degenerate scale. We study the nonunique solution analytically for the circular plate and find degenerate scales. The similar properties of solvability condition between the membrane (Laplace) and plate (biharmonic) problems are also examined. The number of degenerate scales for the six boundary integral formulations is also determined. Tel.: 886-2-2462-2192-ext. 6140 or 6177     

Scattering from two eccentric spheroids: Theory and numerical investigation

In this paper, the direct acoustic scattering problem of a point source field by a penetrable spheroidal scatterer hosting an impenetrable spheroidal body of arbitrary position, size and orientation, is considered. The application background corresponds to the near field measurement of the acoustic field, scattered by a soft-tissue organ including a hard inhomogeneity. The methodology incorporates two independent techniques which are modified appropriately to fit together and are combined for the first time: first, the Vekua method, which is based on the well known Vekua transformation, providing with fully analytic solutions of Helmholtz equation and second, the method of auxiliary sources in order to represent the net wave contribution of the inhomogeneity. The satisfaction of transmission and boundary conditions is accomplished via the collocation method while the wave character of the fields and the outwards propagating property of the exterior wave are implicitly guaranteed in exact form through the analytic nature of the method. Special effort has been devoted to the self-evaluation of the method by constructing and calculating an indicative error function representing the failure of satisfaction of the boundary conditions on a rich grid over the interfaces, much larger than the set of collocation points, where the error is by construction negligible. This numerical approach leads to very reliable results. The determination of the near scattered field as well as of the far-field pattern are the final outcomes of the present work, providing a thorough solution of the direct scattering problem and giving insight to the corresponding inverse problem.     

On the null and nonzero fields for true and spurious eigenvalues of annular and confocal elliptical membranes

In this paper, the dual boundary element method (BEM) and the null-field boundary integral equation method (BIEM) are both employed to solve two-dimensional eigenproblems. The positions of true and spurious eigenvalues for circular, elliptical, annular and confocal elliptical membranes are analytically examined in the continuous system and numerically studied in the discrete system. To analytically study eigenproblems, the polar and elliptical coordinates in conjunction with the Bessel functions, the Mathieu functions, the Fourier series and eigenfunction expansions are adopted. The fundamental solution is expanded into the degenerate kernel while the boundary densities of circular and elliptical boundaries are expanded by using the Fourier series and eigenfunction expansion, respectively. Dirichlet and Neumann eigenproblems are both considered as well as simply and doubly-connected domains are both addressed. By employing the singular value decomposition (SVD) technique in the discrete system, the common right unitary vectors corresponding to the true eigenvalues for the singular and hypersingular formulations are found while the common left unitary vectors corresponding to the spurious eigenvalues are obtained for the singular formulation or hypersingular formulation. True eigenvalues depend on the boundary condition while spurious eigenvalues depend on the approach, the singular formulation or hypersingular formulation of BEM/BIEM. Nonzero field in the domain are analytically derived and are numerically verified in case of the true eigenvalue while the interior null field and nonzero field for the complementary domain are obtained in case of the spurious eigenvalue. Four examples, circular, elliptical, annular and confocal elliptical membranes, are considered to demonstrate the finding of the present paper. After comparing with the analytical and numerical results, good agreements are made. The dual BEM displays the dual structure in the unitary vector and the null field.     

Rotary oscillations of a spheroid in an incompressible micropolar fluid

The paper discusses the flow generated by rotary oscillations of a spheroid (prolate and oblate) in incompressible micropolar fluid. The velocity and microrotation components are determined explicitly in terms of spheroidal wave functions and are expressed in infinite series form. The couple on the oscillating spheroid is evaluated and numerical studies are undertaken to examine the effects of the geometric parameter and material constant parameters of the fluid.     

Internal, near-surface, and scattered electromagnetic fields for a layered spheroid with arbitrary illumination

A spheroidal coordinate separation-of-variables solution has been developed for the determination of internal, near-surface, and scattered electromagnetic fields of a layered spheroid (either prolate or oblate) with arbitrary monochromatic illumination (e.g., plane wave or focused Gaussian beam). Calculated results are presented for layered 2:1 axis ratio prolate and oblate spheroids with an equivalent sphere size parameter of 20.     

Ultrasonic Wave Scattering by a Line Crack in a Solder Joint

A numerical procedure for the solution of a wave propagation problem in a solder joint with a line crack in its base layer is presented. The two-dimensional ``in-plane' wave propagation problem for a finite, multilayered body with a line crack in one of the layers is solved by the combined usage of the displacement and the traction BIEM. The discretization of the boundary with parabolic elements far from the crack edge and with quarter-point crack-tip boundary elements containing the correct behavior for displacement variations at the crack edge is used. Numerical results for a solder joint with real geometry and physical properties are presented. The relations between the wave scattering problems, the solder joint fatigue state estimation, and the reliability and quality of electronic packages is discussed.     

Scattering of electromagnetic radiation by multilayered spheroidal particles: recursive procedure

A formalism is developed for the calculation of the electromagnetic field scattered by a multilayered spheroidal particle. The suggested formalism utilizes the recursive approach with respect to passing from one layer to the next; thus it does not require an increase in the size of the equation matrices involved as the number of layers increases. The equations operate with matrices of the same size as for a homogeneous spheroid. The special cases of extremely prolate and weakly prolate spheroids are considered in more detail. It is shown that in such cases one can avoid the matrix calculations by instead using the iterative scalar calculations.     

Dielectric Response of a Charged Prolate Spheroid in an Electrolyte Solution

We derived the so-called standard set of electrokinetic equations in prolate spheroidal coordinates for all ionic strengths, zeta potentials, and applied electric field frequencies, with the assumption, however, that the particle’s electrophoretic mobility is small. We subsequently solved these equations using finite differences methods. We show that the dipolar coefficient of a prolate spheroid reduces to that of a sphere in the corresponding limit, but deviates strongly from it when the eccentricity of the spheroid is large, for the same particle volume. We also verified that a previously published analytical theory (Chassagne and Bedeaux, J. Colloid Interface Sci., 326:240, 2008) is in good agreement with the numerical results for a large range of zeta potentials, ionic strengths, and frequencies.     

Ground state systematics and collective dipole excitations of spheroidally deformed sodium clusters in a diffuse jellium model

We solve the Kohn-Sham equations self-consistently in the local density approximation for spheroidal sodium clusters in the particle range 8 ≤ Z ≤ 40. We use a smooth fermi-like jellium density to simulate the influence of the ions in the surface region and obtain similar results as Ekardt and Penzar, but slightly different regions of prolate-to-oblate transitions. We present the systematics of potential energy curves with respect to transitions between oblate, prolate and spherical shapes. Shape transitions occur at particle numbers 12/14 (prolate/oblate), 18/20/22 (oblate/spherical/prolate) and 30/32 (prolate/oblate), which are in good agreement with experimental results. The quadrupole and hexadecupole overlap of the electron density with the jellium is investigated, showing a strong hexadecupole dependence for selected clusters. Collective dipole resonances are described in a simple sum rule approach, which reveals a double splitting according to the different resonance frequencies along the principal axes of the spheroid. The systematics of the resonance peaks for the larger clusters with Z ≥ 20 is in good agreement with experimental results.     

Stokes flow past a swarm of porous approximately spheroidal particles with Kuwabara boundary condition

The solution of the problem of symmetrical creeping flow of an incompressible viscous fluid past a swarm of porous approximately spheroidal particles with Kuwabara boundary condition is investigated. The Brinkman equation for the flow inside the porous region and the Stokes equation for the outside region in their stream function formulations are used. As boundary conditions, continuity of velocity and surface stresses across the porous surface and Kuwabara boundary condition on the cell surface are employed. Explicit expressions are investigated for both inside and outside flow fields to the first order in a small parameter characterizing the deformation. As a particular case, the flow past a swarm of porous oblate spheroidal particles is considered and the drag force experienced by each porous oblate spheroid in a cell is evaluated. The dependence of the drag coefficient on permeability for a porous oblate spheroid in an unbounded medium and for a solid oblate spheroid in a cell on the solid volume fraction is discussed numerically an and graphically for various values of the deformation parameter. The earlier known results are then also deduced from the present analysis.