There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

The electric field integral equation (EFIE) is used with the moment method to develop a simple and efficient numerical procedure for treating problems of scattering by arbitrarily shaped objects. For numerical purposes, the objects are modeled using planar triangular surfaces patches. Because the EFIE formulation is used, the procedure is applicable to both open and closed surfaces. Crucial to the numerical formulation is the development of a set of special subdomain-type basis functions which are defined on pairs of adjacent triangular patches and yield a current representation free of line or point charges at subdomain boundaries. The method is applied to the scattering problems of a plane wave illuminated flat square plate, bent square plate, circular disk, and sphere. Excellent correspondence between the surface current computed via the present method and that obtained via earlier approaches or exact formulations is demonstrated in each case.

Electromagnetic scattering by surfaces of arbitrary shape

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Table Of Integrals Series And Products

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

: The book represents an exhaustive study of the scattering properties of acoustically soft and hard bodies and of perfect conductors, presented for 15 geometrically-simple shapes Such shapes are important in their own right and as a basis for synthesizing the radiation and scattering properties of more complex configurations Each shape is treated in a separate chapter whose contents are presented in stylized format for easy reference Emphasis is placed on results in the form of formulae and diagrams Although no detailed derivation are included, an outline of methods in scattering theory is given in the Introduction

Electromagnetic and acoustic scattering by simple shapes

A new method for computing the frequency-domain electromagnetic fields scattered from, and penetrating into, arbitrarily shaped, three-dimensional, lossy, inhomogeneous anisotropic scatterers is presented. The method is based on a general volume integro-differential formulation of the scattering problem, and consists of the numerical solution of the coupled integral equations by the moment method and point matching. A particularly powerful feature of this method is that the numerical model of the scatterer is obtained by parametric volume elements and the basis functions used to represent the field within each element are the same used in the finite-element method. Element integration problems due to the singular kernel of the integral equations are treated in some detail. Numerical results for both the isotropic and the anisotropic spherical scatterer are presented, including comparisons with results obtained by different numerical methods for the isotropic cases considered. The capability of the numerical code presented here to deal with cases where the material parameters of the scatterer are given by singular matrices is discussed for two particular examples. >

Moment method with isoparametric elements for three-dimensional anisotropic scatterers

Integro-differential equations are obtained for the electric and magnetic fields inside a linear, lossy, and anisotropic scatterer, in the frequency domain. The material of the scatterer is characterized by arbitrary values of the elements of the relative permittivity tensor \bar{\epsilon} , the relative permeability tensor \bar{\mu} , and the conductivity tensor \bar{\sigma} . The results are specialized to the case of oblique scattering in two dimensions, for which a numerically efficient computer code has been developed and tested, as described in a companion paper (Part II). The method developed herein is based on a comparison between macroscopic and microscopic descriptions of electromagnetic fields in material media, and is of general applicability.

Electromagnetic scattering from anisotropic materials, part I: General theory

The dispersion relation for an arbitrary general bianisotropic medium is derived in Cartesian coordinates, in a form well suited to imposing the boundary conditions when dealing with layered media with planar and parallel interfaces. Special cases of practical interest are also considered. Eleven fundamental coefficient families are identified by considering in detail all the symmetries present in the dispersion relation. An ad hoc expression of the determinant of the sum of two 3*3 matrices permits the use of a simple procedure to obtain the coefficients of the dispersion equation. The discussed symmetry properties have general validity, and this technique to evaluate the coefficients may be useful in other fields of application where dispersion relations are of importance. >

Dispersion relation for bianisotropic materials and its symmetry properties

Theory and measurements for the determination of the constitutive parameters of an anisotropic material are described, when a slab of the material is inserted in a rectangular waveguide. If both epsilon and µ tensors have zero off-diagonal elements (biaxial material), then the six diagonal elements can be determined by measuring amplitude and phase of reflection and transmission coefficients. If the material is nondispersive, two sets of measurements at two different frequencies are sufficient, under TE/sub 10/ excitation. In the more general case of a lossy and dispersive material, two sets of measurements at the same frequency under TE/sub10/ and TE/sub20/ excitations are needed. An experimental setup for the latter case is described.

Piergiorgio L. E. Uslenghi

Papers

Electromagnetic and acoustic scattering by simple shapes

Moment method with isoparametric elements for three-dimensional anisotropic scatterers

Electromagnetic scattering from anisotropic materials, part I: General theory

Dispersion relation for bianisotropic materials and its symmetry properties

The Inverse Problem for Biaxial Materials