Terms Of Surface Waves Research Articles

Propagation of waves along an impedance boundary

A theoretical analysis of the scalar wave field due to a point source above a plane impedance boundary is presented. A surface wave is found to be an essential component of the total wave field. It is shown that, as a result of ducting of energy by the surface wave, the amplitude of the total wave near the boundary can be greater than it would be if the boundary were perfectly reflecting. Asymptotic results, valid near the boundary, are obtained both for the case of finite impedance (the soft-boundary case) and for the limiting case in which the impedance becomes infinite (the hard-boundary case). In the latter, the wave amplitude in the farfield decreases essentially as r−1, where r is the horizontal propagation distance; whereas in the former (if the surface-wave term is neglected) it decreases as r−2. The results obtained here are compared with results of experimental studies of sound propagation along boundaries, and it is suggested that some apparently anomalous experimental findings might be explained in terms of the present theory.

Radio Wave Propagation in Stratified Media with Nonuniform Boundaries and Varying Electromagnetic Parameters: Full Wave Analysis

Full wave solutions are derived to the problem of radio wave propagation in stratified media with non-uniform boundaries and varying electromagnetic parameters. The analysis employs a complete set of forward and backward travelling waves. The continuous parts of the wavenumber spectrum constitute the radiation field and the lateral waves, and the discrete part of the spectrum is identified as the surface wave term. The solutions are not restricted by the approximate surface impedance concept and the source and receiver can be situated on opposite sides of the interface. Exact boundary conditions are imposed and the solutions are shown to be consistent with the reciprocity relationships.

Anomalous Loss at a Ferrite Boundary

The occurrence of anomalous loss and its explanation in terms of surface waves is discussed. For this type of explanation to be possible the region of occurrence of the surface wave must at least straddle the region of anomalous loss. It is shown that this is so, particularly for the case when there is a mixed air-ferrite surface layer for which this result is not obvious: as the air content decreases, a ferrite-metal surface wave appears and takes over the function of the layer wave. The means by which these waves are generated, and the determination of their amplitudes, appear to require a new physical principle to be applied. A new type of "edge condition" is postulated.

Nuclear glory scattering

A simple model in terms of surface waves for nuclear back-scattering analogous to the optical explanation of the meteorological glory is derived for spin-0 and spin- 1 2 particles from spin-0 targets. Numerous examples of α scattering from spinless nuclei in the energy range from 18 to 50 MeV and π + scattering from protons in the range 2 to 8 GeV are fit successfully with the model. The model yields estimates of the radius of interaction whose values are in agreement with radii determined by other means. Other theories and models relating to backscattering are discussed and attention is called to the similarity between our model and the Regge-pole treatment of this problem by Ceuleneer, Demeur, and Reignier.

Creeping-Wave Description in Sound Scattering from Elastic Cylinders

Plane-wave sound scattering from infinite cylinders results in creeping waves for a rigid or pressure release surface, and in circumferential waves for elastic cylinders (which may include creeping waves altered by elastic effects). Calculations such as backscattering cross sections for rigid or soft cylinders are greatly facilitated by the creeping-wave method, which involves transforming the normal-mode solution by the Sommerfeld-Watson transformation. The physical interpretation in terms of surface waves made possible by this method suggests that the method yields a meaningful description of circumferential waves in elastic shells. The search for such waves is reduced by the creeping-wave method to the location of roots of a 6×6 determinant or, for a solid cylinder, a 3×3 determinant, involving cylindrical functions of complex order.

High-frequency scattering by an impenetrable sphere

The scattering of a scalar plane wave by a totally reflecting sphere (hardcore potential) at high frequencies is treated by a modified Watson transformation. The behavior of the solution both in the near and far regions of space is discussed, as well as the accuracy and domain of applicability of the WKB approximation and classical diffraction theory. It is shown that different transformations are required in the forward and backward half-spaces, and corresponding integral representations for the primary wave are derived. The transformations are rigorously proved and the convergence of the residue series is discussed. In the shadow region, the physical interpretation of the complex angular momentum poles in terms of surface waves is in agreement with Keller's geometrical theory of diffraction. In the lit region, sufficiently far from the shadow boundary, the WKB expansion for the wave function is confirmed up to the second order. On the surface of the sphere, Kirchhoff's approximation is accurate, except in the penumbra region, where the behavior is described by Fock's function. The diffraction effects in the neighborhood of the shadow boundary are investigated and the corrections to classical diffraction theory are obtained. The shift of the shadow boundary is evaluated. The expression for the wave function in the Fresnel-Lommel region is derived and applied to the discussion of the Poisson spot and the behavior near the axis. The total scattering amplitude is evaluated for all angles, including the neighborhood of the forward and backward directions. The corrections to the forward diffraction peak and the transition to the region of geometrical reflection are discussed. The modified Watson transformation is also applied directly to the scattering amplitude. The connection between representations valid in different regions is established.

On a method to obtain a Green's function for a multi-layered half space

Abstract In this paper the surface wave terms of the Green's function for a two-dimensional multilayered half space are obtained. The method used is new and remarkable by its simplicity. It is based on the integral representation theorems for elastodynamics. The orthogonality properties of surface waves are generalized to include not only Love waves but Rayleigh waves as well.

On the excitation of the waves of proper solutions

An exact and explicit solution is obtained for an electro-magnetic field excited by a vertical electric dipole located over a flat plane of arbitrary surface impedance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z</tex> . Though there are four asymptotic expansions, the expansion by Hadamard's method seems to have the most explicit physical meaning. According to this expansion only the proper solution of the ordinary surface wave can be present explicitly on the established condition of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">arg(Z)&gt;\pi/4</tex> . The variation of the field on a spherical surface due to the change of surface impedance is investigated by the use of the ordinary B. Van der Pol and H. Bremmer formula. In this case, when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\arg(Z) &gt;\pi/3</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">|Z|</tex> is larger than some definite value <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">|Z_{0}|</tex> , one of the terms of the formula is found to have the asymptotic form similar to the surface wave term in a flat surface case and, when the surface is highly inductive, it becomes the leading term at large distance from the dipole. The height-gain factor of this term first decreases rapidly with the height up to some point and then gradually increases. The growing up at large height is due to the radiation field. On the other hand, when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">|Z|\ll|Z_{0}|</tex> , the leading term can scarcely have the correspondence with that in a flat surface case. The propagation of surface wave over a spherical surface across several boundaries of discontinuity of the surface impedance is considered. The leading term is found to take the simple form as in the ordinary transmission line. The results in the case of flat plane are derived from these results as the asymptotic forms in the limit of infinite radius of curvature of spherical surface. The physical meaning of the proper solution such as ordinary surface wave is discussed. The physical condition is not the support condition but rather the condition on which, in some domain of space, the proper solution becomes the leading term of wave by an excitation whose dimension is of the order of magnitude of the wavelength or smaller. As in the case of spherical surface, the radiation field and the surface wave field are generally inseparable. The physical wave will be the leading term of the excited wave itself which may sometimes be a proper solution, such as ordinary surface wave, and sometimes be a mixture with the radiation field. The set of proper solutions is not unique. The Green function of the electromagnetic field constitutes the infinite sets of proper solutions of a continuous spectrum, by each set of which the observed field at a given point can be represented, and in each set of which some members of solutions have the properties of surface wave if there is a highly inductive surface. The field strength excited by a dipole is investigated as a function of the surface impedance in the case where the surface is a fiat plane (Section I) or a spherical surface (Section II). The results in the latter case are then applied to the surface wave propagating across several boundaries of discontinuity of surface impedance (Section III). The physical meaning of the waves of proper solutions, such as the surface wave, is finally discussed in Section IV.

The calculation of field strengths over a spherical Earth

Curves and formulae are given for the calculation of field strength at any height and distance from the transmitter for the case of horizontally-polarized electromagnetic waves over a curved Earth or sea. Sufficiently within the optical range the field is calculated by ray theory, and appropriate quantities for calculating path difference and divergence of reflected ray are given. At exact optical cut-off the field is calculated by an approximate method believed to be accurate to 1 db, and is presented in the form of contour curves of equal field strength on a plot of receiver height against transmitter height. Sufficiently far beyond optical cut-off the field is given by the first term in the diffraction-formula series. To calculate the field near optical cut-off, a curve should be drawn of field against distance for the heights of transmitter and receiver in question which joins smoothly the ray-theory curve for inside optical range to the one-term curve for the region beyond and passes through the point determined for the actual cut-off distance. The case of vertically polarized waves is also briefly discussed, and curves and formulae are given for the reflection coefficient (including surface-wave term) where ray theory is applicable, and for one term of the diffraction formula in the region well beyond optical cut-off.

The calculation of field strengths over a spherical Earth

Curves and formulae are given for the calculation of field strength at any height and distance from the transmitter for the case of horizontally-polarized electromagnetic waves over a curved Earth or sea. Sufficiently within the optical range the field is calculated by ray theory, and appropriate quantities for calculating path difference and divergence of reflected ray are given. At exact optical cut-off the field is calculated by an approximate method believed to be accurate to 1 db, and is presented in the form of contour curves of equal field strength on a plot of receiver height against transmitter height. Sufficiently far beyond optical cut-off the field is given by the first term in the diffraction-formula series. To calculate the field near optical cut-off, a curve should be drawn of field against distance for the heights of transmitter and receiver in question which joins smoothly the ray-theory curve for inside optical range to the one-term curve for the region beyond and passes through the point determined for the actual cut-off distance. The case of vertically polarized waves is also briefly discussed, and curves and formulae are given for the reflection coefficient (including surface-wave term) where ray theory is applicable, and for one term of the diffraction formula in the region well beyond optical cut-off.