High-frequency refraction and diffraction in general media

Abstract

The working hypothesis of this paper is that the effect of an opaque boundary on the propagation of high-frequency waves in a general medium is to produce a wave reflected according to the laws of geometrical optics together with a field which to a first approximation depends upon the difference between the curvatures of a tangent ray and the boundary. In order to determine the latter field the model of a medium, whose properties vary linearly, above a straight boundary is employed. A first approximation to the field with this model is found, together with an estimate of the error. The formula for the field is then cast into a form which is invariant under a conformal mapping. Since the difference in curvatures of a tangent ray and the boundary is invariant it is suggested that the field is applicable for all media and boundaries provided that certain conditions imposed in deriving the approximation are fulfilled. As a check the predictions of the formula are compared with independent calculations on (i) a stratified medium above a straight boundary, (ii) a circular cylinder in a homogeneous medium, (iii) a parabolic cylinder in a homogeneous medium, (iv) a circular cylinder in a circularly stratified medium. In all cases the two calculations are in agreement. In a final section the results are extended to phenomena which are aperiodic in time. The proposed universal formula is simple to apply, requiring only the calculation of rays in the medium.