A. Charalambopoulos

An analytic solution for low-frequency scattering by two soft spheres

A plane wave is scattered by two small spheres of not necessarily equal radii. Low-frequency theory reduces this scattering problem to a sequence of potential problems which can be solved iteratively. It is shown that there exists exactly one bispherical coordinate system that fits the given geometry. Then R-separation is utilized to solve analytically the potential problems governing the leading two low-frequency approximations. It is shown that the Rayleigh approximation is azimuthal independent, while the first-order approximation involves the azimuthal angle explicitly. The leading two nonvanishing approximations of the normalized scattering amplitude as well as the scattering cross-section are also provided. The Rayleigh approximations for the amplitude and for the cross-section involve only a monopole term, while their next order approximations are expressed in terms of a monopole as well as a dipole term. The dipole term disappears whenever the two spheres become equal, and this observation provide...

Electromagnetic scattering by a triaxial homogeneous penetrable ellipsoid: Low‐frequency derivation and testing of the localized nonlinear approximation

The field resulting from the illumination by a localized time-harmonic low-frequency source (typically a magnetic dipole) of a voluminous lossy dielectric body placed in a lossy dielectric embedding is determined within the framework of the localized nonlinear approximation by means of a low-frequency Rayleigh analysis. It is sketched (1) how one derives a low-frequency series expansion in positive integral powers of (jk), where k is the embedding complex wavenumber, of the depolarization dyad that relates the background electric field to the total electric field inside the body; (2) how this expansion is used to determine the magnetic field resulting outside the body and how the corresponding series expansion of this field, up to the power 5 in (jk), follows once the series expansion of the incident electric field in the body volume is known up to the same power; and (3) how the needed nonzero coefficients of the depolarization dyad (up to the power 3 in (jk)) are obtained, for a general triaxial ellipsoid and after careful reduction for the geometrically degenerate geometries, with the help of the elliptical harmonic theory. Numerical results obtained by this hybrid low-frequency approach illustrate its capability to provide accurate magnetic fields at low computational cost, in particular, in comparison with a general purpose method-of-moments code.

The localized nonlinear approximation in ellipsoidal geometry: a novel approach to the low-frequency scattering problem

The localized nonlinear approximation provides a very effective method within the integral equation framework of electromagnetic scattering theory. Existing results in this direction are confined to the spherical geometry alone. In this work, we extend the known results for the sphere to the case of ellipsoidal geometry which can approximate genuine three-dimensional scattering obstacles. Reduction to prolate and oblate spheroids, where rotational symmetry is present, is also discussed.

Complete decomposition of axisymmetric Stokes flow

Abstract As Stokes has shown, axisymmetric, incompressible, viscous creeping flow can be studied through the use of a stream function Ψ which belongs to the kernel of the fourth-order differential operator E 4 , where E 2 Ψ measures the vorticity of the flow. In fact, irrotational flows are described by stream functions that belong to the ker E 2 , while rotational flows are described by stream functions that do not belong to the ker E 2 . It is shown that a decomposition, of the form Ψ = Ψ 1 + r 2 Ψ 2 , for any stream function Ψ is possible, where Ψ 1 and Ψ 2 belong to ker E 2 and r is the radial spherical variable. Consequently, a stream function that describes a rotational flow can always be divided by a stream function that describes an irrotational flow in a way that renders the ratio always equal to the square of the Euclidean distance. If no singularities are observed on the axis of symmetry then the above decomposition is unique.

Free vibrations of the viscoelastic human skull

In this work we deal with the free vibrations of a viscoelastic skull. The analysis is based on the three-dimensional theory of viscoelasticity and the representation of the displacement field in terms of the Navier eigenvectors. The frequency equation was solved numerically and results are presented for the eigenfrequency and the attenuation spectra.

Free vibrations of a double layered elastic isotropic cylindrical rod

In this work we propose an approach to study the dynamic characteristics of double layered cylindrical rods. The description of the problem is based on the three-dimensional theory of elasticity and the mathematical analysis on the representation of the displacement fields in terms of the constructed Navier eigenvectors in cylindrical coordinates. Finally, for a special case, the proposed analysis is presented in detail and the results obtained are in excellent agreement with the existing ones.

Scattering of a point generated field by kidney stones

SummaryIn this work, we examine the acoustic scattering problem of spherical waves by a two-layer spheroid simulating the kidney-stone system. Both the theoretical as well as the numerical treatment are presented. The outcome of the analysis is the determination of the scattered field along with its multivariable dependence on the several physical and geometric parameters of the system. A comparison with the simpler case of spherical geometry as well as the plane wave excitation case is realised.

The effect of geometry on the dynamic characteristics of the human skull

In the present work we discuss the role of small deviation of the spherical to spheroidal geometry on the frequency spectrum of the human dry-skull. The analysis is based on the three dimensional theory of elasticity, complex analysis techniques and the construction of the Navier eigenvectors for the problem under discussion.

Note on eigenvector solutions of the Navier equation in cylindrical coordinates

SummaryThe eigenvector solution of the spectral Navier equation in cylindrical coordinates is constructed by using the Helmholtz decomposition theorem and the method of separation of variables.

The effect of viscoelastic brain on the dynamic characteristics of the human skull-brain system

SummaryIn this paper, we present a solution to the problem of free vibrations of the human head system taking into account the dissipative behaviour of the brain. The mathematical model is based on the three-dimensional theory of viscoelasticity and the representation of the displacement field in terms of the Navier eigenvectors. The frequency equation is solved numerically and the results for eigenfrequencies and damping coefficients are presented for various geometrical and physical parameters of the system. The results obtained are in excellent agreement with the measured eigenfrequencies, and the predicted damping coefficients are within the order of magnitude of the measured ones. From the proposed analysis we have obtained the information that the role of the viscoelastic neck as well as the viscoelastic properties of the skull-brain system have to be simultaneously taken into account in the study of the frequency spectrum of the human head. The analysis of the realistic model is under preparation.