In the study of short-wave diffraction problems, asymptotic methods - the ray method, the parabolic equation method, and its further development as the "etalon" (model) problem method - play an important role. These are the meth ods to be treated in this book. The applications of asymptotic methods in the theory of wave phenomena are still far from being exhausted, and we hope that the techniques set forth here will help in solving a number of problems of interest in acoustics, geophysics, the physics of electromagnetic waves, and perhaps in quantum mechanics. In addition, the book may be of use to the mathematician interested in contemporary problems of mathematical physics. Each chapter has been annotated. These notes give a brief history of the problem and cite references dealing with the content of that particular chapter. The main text mentions only those pUblications that explain a given argument or a specific calculation. In an effort to save work for the reader who is interested in only some of the problems considered in this book, we have included a flow chart indicating the interdependence of chapters and sections.
Table of Contents1. Introduction.- 2. The Ray Method.- 2.1 The Basic Principles.- 2.2 Variational Theory of the Fermat Functional.- 2.3 The Solution of the Eikonal Equation; Ray Coordinates and the Geometrical Divergence.- 2.4 Integration of the Transport Equations.- 2.5 Maxwell’s Equations.- 2.6 Determining the Short-Wavelength Asymptotic Solution of a Diffraction Problem Using the Ray Method – An Example.- 2.7 Determination of the Function ?0 by Using the Localization Principle.- 2.8 Caustics.- 2.9 Notes on the Literature.- 3. The Field Near a Caustic.- 3.1 Preliminary Remarks.- 3.2 The Etalon Problem for Caustics.- 3.3 The Ray Field and Eikonal in the Neighborhood of a Caustic.- 3.4 Derivation of the Recurrence Relations.- 3.5 The Field in the Vicinity of a Caustic – First Approximation.- 3.6 Determination of Aj and Bj for j > 0.- 3.7 Determination of the ?j.- 3.8 Notes on the Literature.- 4. Derivation of Asymptotic Formulas for Eigenvalues and Eigenfunctions Using the Ray Method.- 4.1 Introductory Remarks.- 4.2 Multi-Sheeted Covering Spaces.- 4.3 Single-Valuedness of the Eigenfunctions and Quantization Conditions.- 4.4 Eigenvalues and Eigenfunctions of a Circle.- 4.5 Eigenvalues of an Ellipse.- 4.6 Notes on the Literature.- 5. The Ray Method “in the Small”.- 5.1 Eigenfunctions of the Whispering Gallery Type.- 5.2 Eigenvalues of the Bouncing Ball Type.- 5.3 Eigenvalues of the Whispering Gallery Type for a Nonconstant Wave Velocity.- 5.4 Eigenvalues of the Bouncing Ball Type for a Nonconstant Wave Velocity.- 5.5 Notes on the Literature.- 6. The Parabolic Equation Method.- 6.1 Introductory Remarks.- 6.2 Derivation of the Parabolic Equation for Eigenfunctions of the Whispering Gallery Type.- 6.3 Solution of the Parabolic Equation (6.2.9); Asymptotic Expansion of Eigenfunctions of the Whispering Gallery Type.- 6.4 Derivation of the Basic Parabolic Equation for the Case Where S Is a Ray.- 6.5 Solution of the Parabolic Equation (6.4.8).- 6.6 Notes on the Literature.- 7. Asymptotic Expansions of Eigenfunctions Concentrated Close to the Boundary of a Region.- 7.1 Introductory Remarks.- 7.2 Eigenfunctions of the Circle for the Case c = const.- 7.3 Construction of Solutions of the Helmholtz Equation in a Boundary Layer.- 7.4 Eigenfunctions of the Whispering Gallery Type.- 7.5 Eigenfunctions of the Region Exterior to ?.- 7.6 Justification of the Asymptotic Formulas.- 7.7 Notes on the Literature.- 8. Eigenfunctions Concentrated in the Neighborhood of an Extremal Ray of a Region.- 8.1 The Etalon Problem.- 8.2 Construction of the Principal Terms of the Formal Series.- 8.3 Construction of the Polynomials ?m and ?m, m ? 1.- 8.4 Basic Results and Some of Their Consequences.- 8.5 Formulation of the Boundary Value Problem and Derivation of the Eigenvalue Equation.- 8.6 Formulas for Eigenvalues and Eigenfunctions in the First Approximation.- 8.7 Procedure for Constructing the Polynomials ?m(s, ?) and ?m(s, ?) for m ? 1.- 8.8 Natural Frequencies of an Open Resonator (Inhomogeneous Filling, Higher Approximations).- 8.9 Notes on the Literature.- 9. Eigenfunctions Concentrated in the Vicinity of a Closed Geodesic.- 9.1 Formulation of the Problem and Derivation of the Parabolic Equation.- 9.2 The Jacobi Equation for the Geodesic l.- 9.3 The Zero-Order Approximation.- 9.4 Construction of the Higher Approximations.- 9.5 The Eigenfunction Problem in a Three-Dimensional Region.- 9.6 Asymptotic Solution of a System of Elliptic Equations on a Riemannian Manifold, Concentrated Near a Ray.- 9.7 Notes on the Literature.- 10. Multiple-Mirror Resonators.- 10.1 The Multiple-Mirror Resonator and Formulation of the Problem.- 10.2 Conditions of Resonator Stability in the First Approximation.- 10.3 Some Properties of the Solutions of (10.2.16) on lN.- 10.4 Formulation of the Parabolic Equation for the Problem.- 10.5 Integration of the Equation LV = 0.- 10.6 Eigenfunctions and Natural Frequencies of a Multiple-Mirror Resonator in the First Approximation.- 10.7 Construction of the Higher Approximations.- 10.8 Notes on the Literature.- 11. The Field of a Point Source Located Near a Convex Curve.- 11.1 Introduction.- 11.2 The Green’s Function for the Exterior of a Circle.- 11.3 Creeping Waves Near a Curve with Positive Curvature and Their Extension to Arbitrary Distances.- 11.4 An Expression for the Green’s Function in Terms of Creeping Waves.- 11.5 The Green’s Function for the Diffraction Problem at a Cylinder with Variable Impedance.- 11.6 Notes on the Literature.- 12. Asymptotic Expansion of the Green’s Function for a Surface Source (the Internal Problem).- 12.1 Formulation of the Problem and Physical Assumptions.- 12.2 The Ray Formula for Multiply Reflected Waves.- 12.3 Refinement of the Ray Formula.- 12.4 Field of a Source Located on the Boundary of a Circle.- 12.5 Field of a Surface Source Close to the Concave Boundary of an Inhomogeneous Body.- 12.6 Notes on the Literature.- 13. The High-Frequency Asymptotics of the Field Scattered by a Smooth Body.- 13.1 The Etalon Problem.- 13.2 Construction of Approximate Caustic Sums – Equations for the Expansion Coefficients.- 13.3 Asymptotic Evaluation of the Integral I1, in the Vicinity of the Terminator C.- 13.4 Choice of the Initial Data; Fock’s Formula.- 13.5 Transformation of the Integrals I1 and I2 in the Neighborhood of the Light-Shadow Boundary.- 13.6 Calculation of the Derivatives of ?+(M, ?) and u+(M, ?) on ?+.- 13.7 The Fresnel-Fock Formula in the Neighborhood of the Light-Shadow Boundary.- 13.8 Asymptotics of the Field in the Deep Shadow.- 13.9 Notes on the Literature.- A.1. The Airy Equation and Airy Functions.- A.2. Nonorthogonal Curvilinear Coordinate Systems.- References.