R. H. Tew

On the Theory of Complex Rays

The article surveys the application of complex-ray theory to the scalar Helmholtz equation in two dimensions. The first objective is to motivate a framework within which complex rays may be used to make predictions about wavefields in a wide variety of geometrical configurations. A crucial ingredient in this framework is the role played by Sp{} in determining the regions of existence of complex rays. The identification of the Stokes surfaces emerges as a key step in the approximation procedure, and this leads to the consideration of the many characterizations of Stokes surfaces, including the adaptation and application of recent developments in exponential asymptotics to the complex Wentzel--Kramers--Brilbuin expansion of these wavefields. Examples are given for several cases of physical importance.

Stokes phenomenon and matched asymptotic expansions

This paper describes the use of matched asymptotic expansions to illuminate the description of functions exhibiting Stokes phenomenon. In particular the approach highlights the way in which the local structure and the possibility of finding Stokes multipliers explicitly depend on the behaviour of the coefficients of the relevant asymptotic expansions.

Scalar wave diffraction by tangent rays

Abstract Modern asymptotic methods are used to provide as complete as possible a description of the scattering of a high-frequency time-harmonic acoustic plane wave by a two-dimensional convex obstacle. The cases of infinite and finite scatterers are considered separately, and descriptions of the diffracted fields and the transition solutions valid across shadow boundaries are presented. For a finite scatterer, expressions are given for the directivity function (or angular variation) of the scattered field in the very far-field encompassing all angular directions; in particular this results in the prediction of a narrow range of angles for which the directivity is an order of magnitude lower than in general.

Multiple scales expansions near critical rays

The theory of geometrical optics breaks down whenever a ray is excited tangentially to a boundary. Examples of this include the total internal reflection of a wave incident upon the boundary between two media and the excitation of a surface wave. Model problems for both phenomena are considered, taking the incident fields to be a modulation of an appropriate plane wave. Using a multiple scales analysis, a description of the local field near the boundary point where the tangential ray is excited is obtained in each case. These analyses clearly indicate the mechanisms by which further rays are radiated from the boundary (forming the head wave and acoustic surface wave, respectively) and predict rapidly varying, nonspecular features in the total reflected field.

Thin-Layer Solutions of the Helmholtz and Related Equations

This paper concerns a certain class of two-dimensional solutions to four generic partial differential equations—the Helmholtz, modified Helmholtz, and convection-diffusion equations, and the heat conduction equation in the frequency domain—and the connections between these equations for this particular class of solutions. Specifically, we consider “thin-layer” solutions, valid in narrow regions across which there is rapid variation, in the singularly perturbed limit as the coefficient of the Laplacian tends to zero. For the well-studied Helmholtz equation, this is the high-frequency limit and the solutions in question underpin the conventional ray theory/WKB approach in that they provide descriptions valid in some of the regions where these classical techniques fail. Examples are caustics, shadow boundaries, whispering gallery, and creeping waves and focusing and bouncing ball modes. It transpires that virtually all such thin-layer models reduce to a class of generalized parabolic wave equations, of which the heat conduction equation is a special case. Moreover, in most situations, we will find that the appropriate parabolic wave equation solutions can be derived as limits of exact solutions of the Helmholtz equation. We also show how reasonably well-understood thin-layer phenomena associated with any one of the four generic equations may translate into less well-known effects associated with the others. In addition, our considerations also shed some light on the relationship between the methods of matched asymptotic, WKB, and multiple-scales expansions.

Exponential asymptotics of a fifth-order partial differential equation

We construct an asymptotic representation for the solution

Non-Specular Reflection Phenomena at a Fluid-Solid Boundary

u(x,t)

Geometrical aspects of the diffraction of critically incident bulk waves and Gaussian beams

of a singularly-perturbed linear fifth-order evolution equation which accounts for the relevant exponentially-small terms in all regions of the complex

Diffraction of sound by a surface inhomogeneity at a fluid-solid interface

x

Acoustical Scattering by a Shallow Surface-Breaking Crack in an Elastic Solid Under Light Fluid Loading

plane. The particular equation that we study is chosen in part to highlight the complexities that arise in high-order examples, resulting in particular from the non-existence of a suitable (steady-state) heteroclinic connection. Key points of this calculation are the identification, location and evolution of the active (in the sense that non-zero, though exponentially-small, terms are switched on across them) Stokes lines, and of the higher-order Stokes lines across which these can be activated or inactivated. In doing so, we need in particular to analyse two ‘levels’ of higher-order Stokes lines and to present the associated mechanisms by which they can themselves be activated or inactivated. By piecing together the information concerning which Stokes lines (both ordinary and higher-order) are active, we are able to deduce systematically which of the competing exponentials that can potentially arise within the asymptotic solution are actually present in each region of the complex plane.