An elementary function that is closely related to the Sommerfeld flat‐earth attenuation function is used to derive an alternative integral equation for propagation over irregular terrain. This choice of elementary function is shown to satisfy a scalar ‘parabolic’ wave equation. For the special case of a spherical earth, the integral equation yields more accurate results than previous general formulations; in addition, the integral equation is numerically feasible for both vertical and horizontal polarization.
Theoretical and experimental studies of the scattering by a two-dimensional periodic array of narrow, perfectly-conducting plates have been carried out. The scattering in the resonance region is treated. The present work is restricted to a normally incident plane wave; however, the approach described here can be extended to the case of oblique incidence.The surfac~ current ], on a single plate is expanded in a series of N terms. An integral equation is obtained for j, by enforcing the boundary condition on the tangential electric field. The N unknown coefficients of}., are found by satisfying the integral equation exactly at N points. The solution for the scattered field was found to be highly sensitive to the location of the N points at which the integral equation is satisfied. A set of N points is found by introducing~ suitable error gauge which involves J }, · ETds. where the integral is taken over a single plate and ET is the calculated total electric field.Within the frequency band considered, the reflectivity of the array is seen to range from unity at its first resonance to zero at certain frequencies where Wood's anomalies occur. Also, the frequency shift of the array resonance from single element resonance is observed. Values of the reflection coef· ficient calculated as a function frequency compare well with experimental values.
An integral equation for cauculating the attenuation of radio waves propagating over irregular terrain is rederived. The integral equation is applied to three terrain profiles, and the solutions are compared with solutions obtained by using classical methods such as the residue series and diffraction theory.
A modification in the mathematical derivation of an integral equation for propagation over irregular, inhomogeneous terrain and a numerically stable algorithm, RING, is developed to extend the frequency range of program WAGNER to frequencies in the ¾HF region. The solution to the integral equation is compared to residue theory and Fock theory and excellent agreement is obtained. the Sommerfeld flat-Earth attenuation function. This integral equation was reduced to a one-dimensional integral equation from the original two-dimensional form by performing a stationary phase integration in the direction transverse to the direction of propagation. The one-dimensional integral equation ignores the effects of scatter perpendicular to the direction of propagation (i.e., side scatter off the great circle path). It does, however, predict the variation in field strength of a radio wave propagating over realistic, smoothly varying, irregular terrain. The source and observer can be located at arbitrary points above or on the terrain profile, and the source can be either horizontally or vertically polarized. This integral equation for propagation over an irregular terrain profile defined by y = y(x) is given by Ott [1971] as f (x) = G[x, y(x)] -f (s)e -ikøø(x's) ß G[x, s; y(x) y(s)] ß [ y' (s) -y(x) -y(s)
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