## Abstract

This paper contains a study of the coupled dynamics of a static, compressible fluid in planar contact with an elastic half-space. The interaction is forced by an incident acoustic bulk wave generated by a uniform, two-dimensional, line source located in the fluid under specified conditions of light fluid loading. An analysis of the exact solution to the corresponding boundary value problem, obtained by Fourier transform methods, is presented and this shows that the ultimate far-field acoustic radiation pattern adopts the usual form of an outgoing cylindrical wave, with an amplitude dependent upon the polar angle $\theta $ and decaying algebraically with distance as R$^{-\frac{1}{2}}$, where (R, $\theta $) are polar coordinates measured from the image source. This is so only for sufficiently large values of R (made precise, in terms of scalings with inverse powers of a small fluid-loading parameter $\epsilon $, in the text) and at lesser distances, a different asymptotic response is found along certain critical directions. This is related to the generation of boundary waves, either of `free-mode' type or else surface disturbances induced by the transmitted elastic waves in the solid. Whatever their specific nature, the corresponding structure is determined by the coalescence (under certain conditions) of a saddle point with an appropriate singularity in the complex wavenumber plane. Each such occurrence is described in some detail and a suitable transitional solution is derived to provide a full and uniform description valid across these critical rays. In particular, a saddle-point-pole coalescence occurs in the course of the analysis of the free-mode type waves (e.g. the leaky Rayleigh wave and the Scholte wave) and it is noted and demonstrated that standard techniques for dealing with this are inadequate, owing to the presence of a simple zero of the integrand under the same conditions. An alternative method for dealing with this non-uniformity is then derived and the implications of this analysis for the acoustic response are discussed. In particular, it is shown that the far-field disturbance in the fluid along the two critical directions of this nature is always of algebraically small O(k$_{0}^{-\frac{1}{2}}$R$^{-\frac{1}{2}}$) amplitude (k$_{0}$ being the acoustic wavenumber), but the precise multiplicative factor changes over a range of distances R = O(k$_{0}^{-1}\epsilon ^{-2}$) with smooth transition afforded by a Fresnel-type integral.

## Royal Society Login

Sign in for Fellows of the Royal Society

Fellows: please access the online journals via the Fellows’ Room

### Log in using your username and password

### Log in through your institution

Pay Per Article - You may access this article or this issue (from the computer you are currently using) for 30 days.

Regain Access - You can regain access to a recent Pay per Article or Pay per Issue purchase if your access period has not yet expired.