Scalar Wave Theory: Green’s Functions and ApplicationsThis 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. |
Contents
1 | |
Contents | 9 |
Appendices | 27 |
in Two Dimensions | 50 |
Appendix | 75 |
Spherical Waves Scattering from Planar Boundaries | 100 |
Appendix | 137 |
Appendix 183 | 182 |
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Common terms and phrases
amplitude analytic angle assume asymptotic Bessel function boundary conditions branch cut branch point c₂ Chap complex constant continuous spectrum contour coordinates cz¹ decay defined delta function density derivative differential equation dimensions eigenfunctions eigenvalues evaluate example expansion exponential finite fluid Fourier transform G₂ Green's function Hankel function Helmholtz equation illustrated in Fig imaginary impedance incident integrand interface K₁ latter normal Note occur one-dimensional outgoing wave P₁ phase term plane wave proper S-L problem quadrant radial real axis reflected field reflection coefficient region representation Retarded Green's Function saddle point satisfies scalar scattering second sheet Sect sin² singularities Snell's law solution sound speed spatial spherical wave square root steepest descent substitute surface tion transmitted field vanishes variable velocity potential wave equation waveguide write written Wronskian yields zero μ₁