The Boundary-Layer Method in Diffraction Problems

It has become almost a cliche to preface one's remarks about asymptotic tech- niques with the statement that only a very few special problems in diffrac- tion theory (be it electromagnetic, acoustic, elastic or other phenomena) are possessed of closed form solutions, but as with many cliches, this is because it is true. One only has to scan the literature to see the large amount of effort (both human and computer) expended to solve diffraction problems involving complicated geometries which do not permit such simplifications as separation of variables, It was a desire for techniques more straightforward than frontal numerical assaults, as well as for a theory \~hich \~ould explain the basic physical phenomena involved, which stimulated research into asymptot- ic methods. Geometrical optics (GO) and, now, even Keller's geometrical theory of dif- fraction (GTD) have been with us for some time, and have become standard tools in the analysis of high-frequency wave phenomena, Of course, it was always recognized that these approaches broke down in certain regions: GO in the shadow region; GTD along shadow boundaries and caustics. One remedy for these defects is to construct an expansion, based upon a more general ansatz than GO or GTD, which is made to be valid in one or more of the areas where GO or GTD break down.

Table of Contents:
Translator's Introduction.- 1 Introduction.- 2 The Ray Method.- 2.1 The Starting Point: Formulas for the Scalar Case.- 2.2 The Eikonal Equation; Rays; Wave Fronts.- 2.3 Ray Coordinates.- 2.4 Fundamental Recurrence Formulas.- 2.5 Reflection of a Wave Given by a Ray Expansion.- 3 The Caustic Problem.- 3.1 Ray Expansion in the Neighborhood of a Caustic.- 3.2 The Analytic Nature of the Eikonal for Incoming and Outgoing Waves Near a Caustic.- 3.3 Ray Series in (s,n) and (s,?) Coordinates.- 3.4 The Field in a Boundary Layer Surrounding the Caustic.- 3.5 Fundamental Formulas.- 4 Whispering Gallery and Creeping Waves.- 4.1 Whispering Gallery Waves.- 4.2 Whispering Gallery Quasimodes.- 4.3 Creeping Waves.- 4.4 The Friedlander-Keller Solution (Diffraction Rays).- 4.5 Matching of Creeping Waves and Diffraction Rays.- 5 Oscillations Concentrated in the Neighborhood of a Ray (Gaussian Beams).- 5.1 Rays in the First Approximation.- 5.2 Derivation of the Boundary-Layer Equation.- 5.3 Solution of the System of Recurrence Equations for Vj.- 5.4 Stability of an Extremal Diameter of a Region.- 5.5 Quasimodes of the "Bouncing-Ball" Type in the First Approximation.- 5.6 Construction of Higher Approximations.- 6 Shortwave Diffraction from a Smooth Convex Body.- 6.1 The Parabolic Equation Method.- 6.2 The Analytic Nature of the Functions $$ {{\rm{\bar V}}_{\rm{j}}} $$ and Vj..- 6.3 The Boundary Layer in the Deep Shadow Zone.- 6.4 Continuation of the Solution from the Vicinity of the Point C into the Transition Region.- 6.5 Analytic Representation of the Incident Wave in the Neighborhood of the Limiting Ray.- 6.6 System of Recurrence Equations for the Neighborhood of the Limiting Ray.- 6.7 Extension of the Transition Region Formulas into the Neighborhood of the Limiting Ray.- 6.8 Formulas for the Field in the Shadow and in the Penumbra.- 7 The Problem of an Oscillating Point Source.- 7.1 The Ray Method for a Central Field of Rays.- 7.2 Expansion in the Transition Region.- 7.3 Expansions in the Neighborhood of the Origin.- 8 Survey of Literature.- References.