The S-Matrix and Its Analysis

This chapter focuses on the scattering matrix, or S-matrix, an infinite-dimensional matrix or operator that expresses the state of a scattering system consisting of waves or particles or both in the far future in terms of its state in the remote past. In the case of electromagnetic (or acoustic) waves, the S-matrix connects the intensity, phase, and polarization of the outgoing waves in the far field at various angles to the direction and polarization of the beam pointed toward an obstacle. The chapter first considers the problem of scattering by a square well, located symmetrically with respect to the origin, before discussing bound states and a heuristic derivation of the Breit-Wigner formula. It als describes the Watson transform and Regge poles before concluding with an analysis of the time-independent radial Schrödinger equation and Levinson's theorem.

Transmission of creeping-wave modes around a periodically modulated impedance cylinder

Motivated by recent advances in radio oceanography, an idealized two-dimensional model of diffraction by a non-uniform cylinder is analyzed. The cylinder is characterized by a surface impedance that varies in a periodic fashion in the azimuthal direction. For excitation by a magnetic line source, a solution for the resulting field is obtained that satisfies the specified periodic boundary condition. This representation for the fields is converted, via a Watson transform procedure, to a form that is more convergent and useful for short wavelengths. It is shown that, for each creeping-wave mode, an infinite number of spatial harmonics is excited. These harmonics are a consequence of the periodic non-uniformity of the surface and they include the backward scattered waves. The latter becomes very strong in the region where the spatial period is half the radio wavelength in accordance with the well-known Bragg phenomenon. In this case, contrary to what is sometimes assumed, the attenuation of the forward creeping waves is much increased. This suggests that perturbation methods, such as used to interpret radio-oceanographic data, may be invalid just in the region when they appear to have the most diagnostic value.

Field Transforms for Multilayered Cylindrical and Spherical Structures of Finite Conductivity

Problems of propagation around multilayered cylindrical or spherical structures possessing highly conducting cores have been analyzed in terms of discrete sets of modes. However, when propagation through the core of the structure is significant and of particular interest, the discrete set of modes is not suitable for the complete expansion of the electromagnetic fields.To provide a suitable basis for the expansion of the electromagnetic field in nonuniform multilayered, cylindrical or spheroidal, dielectric structures we derive expressions for the electric and magnetic field transforms consisting of both a discrete and a continuous spectrum of waves. The relationship between these transforms, the Kontorowich–Lebedev transform, and the discrete Watson transform is discussed. When the radius of curvature of the structure is infinite these transforms merge with generalized Fourier type transforms.The transforms can be used to investigate electromagnetic propagation through irregularly shaped dielectric wave guides, and through irregularly shaped spheroids. It can also be used to solve the problem of propagation in the interior (concave side) of irregularly shaped conducting cylinders or spheroids.

Distributional Watson Transforms

All our notation is as denned in [2] with the restriction to n = 1. However, for our purposes, we introduce a sequence of norms byin It is not difficult to see that turns out to be a fundamental space.It is a well-known fact that the Watson transform and the Mellin transform are connected by the fact thatandif and only if K(s)K(l — s) = 1, where K(s) is the Mellin transform of k(x). Further, the Hankel transform and Hilbert transform can be considered as special cases of Watson transforms.

An earth-flattening transformation for waves from a point source

An earth-flattening transformation is developed for wave-propagation problems that can be formulated in terms of uncoupled scalar Helmholtz equations. Through the transformation, wave problems in isotropic, spherically symmetric media with a specified radial heterogeneity can be expressed in terms of a flat geometry with a suitably vertical heterogeneity. The transformation is exact for homogeneous (no source) problems and is useful for normal mode studies. When a point source of waves is present, the earth-flattening transformation together with the Watson transform converts the reflected wave field from a sum over discrete, spherical eigenfunctions to an integral over continuous wave numbers in a flat geometry. The far-field form of this integral shares many properties with the Weyl integral and is useful for body-wave studies in a spherical earth.