This advanced book will be of interest to researchers working on wave propagation theory and diffraction, in particular in the fields of acoustics, elasticity and geophysics.

1. 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 ofEigenfunctions 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-MirrorResonators.- 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.