Abstract
Abstract Reflection and transmission phenomena associated with high-frequency linear wave incidence on irregular boundaries between adjacent acoustic or electromagnetic media, or upon the irregular free surface of a semi-infinite elastic solid, are studied in two dimensions. Here, an ‘irregular’ boundary is one for which small-scale undulations of an arbitrary profile are superimposed upon an underlying, smooth curve (which also has an arbitrary profile), with the length scale of the perturbation being prescribed in terms of a certain inverse power of the large wave-number of the incoming wave field. Whether or not the incident field has planar or cylindrical wave-fronts, the associated phase in both cases is linear in the wave-number, but the presence of the boundary irregularity implies the necessity of extra terms, involving fractional powers of the wave-number in the phase of the reflected and transmitted fields. It turns out that there is a unique perturbation scaling for which precisely one extra term in the phase is needed and hence for which a description in terms of a Friedlander–Keller ray expansion in the form as originally presented is appropriate, and these define a ‘distinguished’ class of perturbed boundaries and are the subject of the current paper.
Highlights
The problems which are addressed in this paper involve acoustic, electromagnetic, and elastic wave propagation, there are fundamental features common to all three of these situations.First, an oscillatory time-dependence e−iωt is always assumed but suppressed, so that if the wavespeed within the medium in question is c the relevant field equation is the two-dimensional Helmholtz equation ∂2 ∂ x2 + ∂2 ∂ y2 + k2 φ = (1.1)A
The aim is to determine the leading-order solution for both φ r and φt by solving both Helmholtz equations (3.1) and (3.2), subject to the coupled boundary data
Solving the two boundary equations simultaneously whilst ignoring the exponential terms appearing at O (k) and O k1/2 gives the leading-order amplitude of the reflected and transmitted waves along the boundary, which are
Summary
The problems which are addressed in this paper involve acoustic, electromagnetic, and elastic wave propagation, there are fundamental features common to all three of these situations. The variable N will be specified to be N = 2 shortly, but it is helpful to leave N as a general positive integer for the purposes of this introduction In both the acoustic and the electromagnetic cases, the region y < y0(s) + k−1/Ny0(s) is taken to be occupied by a second “lower” medium for which the Helmholtz equation (1.1) is to be replaced by k2 γ2 φ (1.3). The problems have two wave-bearing media both supporting high-frequency, short waves and are separated by a smooth, curved interface superimposed on which are small-scale undulations; the elastic case has the second “lower” medium replaced by a vacuum.
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