This book comprises some of the lecture notes I developed for various one-or two-semester courses I taught at the Colorado School of Mines. The main objective of all the courses was to introduce students to the mathematical aspects of wave theory with a focus on the solution of some specific fundamental problems. These fundamental solutions would then serve as a basis for more complex wave propagation and scattering problems. Although the courses were taught in the mathematics department, the audience was mainly not mathematicians. It consisted of gradu ate science and engineering majors with a varied background in both mathematics and wave theory in general. I believed it was necessary to start from fundamental principles of both advanced applied math ematics as well as wave theory and to develop them both in some detail. The notes reflect this type of development, and I have kept this detail in the text. I believe it essential in technical careers to see this detailed development at least once. This volume consists of five chapters. The first two on Scalar Wave Theory (Chapter 1) and Green's Functions (Chapter 2) are mainly mathematical although in Chapter 1 the wave equation is derived from fundamental physical principles. More complicated problems involving spatially and even temporally varying media are briefly introduced.
Table of Contents1. Scalar Wave Theory.- 1.1 Fluid Equations.- 1.2 Sound.- 1.3 Wave Equations.- 1.4 Spatially Variable Density.- 1.5 Helmholtz Equation and Separable Solutions.- 1.6 One-Dimensional Problems and Sturm-Liouville Theory.- 1.7 Calculation of Eigenvalues from Eigenfunctions.- 1.8 Interface Boundary Conditions.- 1.9 Energy and Momentum.- Appendices.- 1.A Wave Equation Derivation for Spatially Varying Density.- 1 B Wave Equation Derivation for Spatially and Temporally Variable Density with Quadratic Velocity Terms.- 2. Green’s Functions.- 2.1 Basic Definition.- 2.2 Eigenfunction Expansion.- 2.3 One-Dimensional Examples.- 2.4 Elementary Source.- 2.5 Green’s Functions for the Wave Equation.- 2.6 Green’s Functions for the Wave Equation in Two Dimensions.- 2.7 Green’s Functions for the Wave Equation in One Dimension.- 2.8 Green’s Functions for the Helmholtz Equation.- 2.8.1 Three Dimensions.- 2.8.2 Two Dimensions.- 2.8.3 One Dimension.- 2.9 Integral Representations: Helmholtz Green’s Functions.- 2.9.1 Weyl Representation.- 2.9.2 Sommerfeld Representation.- 2.9.3 Explicit Evaluation of G(3).- 2.9.4 Weyrich Representation.- 2.9.5 Plane-Wave Decomposition of G(2).- 2.9.6 Explicit Evaluation of G(2).- 2.9.7 Explicit Relation Between G(3) and G(2).- 2.10 Delta Functions in Different Coordinate Systems.- 2.10.1 Cylindrical Coordinates.- 2.10.2 Spherical Coordinates.- 2.11 Other Examples of Green’s Functions.- 2.A Dirac-Plemelj Relations.- 3. Scalar Plane Wave Scattering.- 3.1 Scattering from a Plane Interface.- 3.2 Examples.- 3.3 Interface Values and Impedances.- 3.4 Impedance Boundary Condition.- 3.5 Wave Fronts and Ray Parameter.- 3.6 Total Transmission.- 3.7 Total Internal Reflection.- 3.8 Effect of Surface Roughness.- 4. Spherical Waves Scattering from Planar c2.- 4.6 Case 2: c1 < c2.- 4.7 Transmitted Field.- 4.A Steepest Descent Method.- 5. Two-Layered Liquid Half-Space (Pekeris Waveguide).- 5.1 Geometry and Notation.- 5.2 Green’s Function.- 5.3 Analytic Properties of G2.- 5.4 Normal Mode (NM) Representation.- 5.5 Fourier-Bessel Representation.- 5.6 Pseudo-proper Modes.- 5.7 Leaky Waves.- 5.8 Branch Line Integral (BLI).- 5.9 Virtual Modes.- 5.10 Modal Attenuation.- 5.A Proof of the Contour Representation.- References.