Scalar wave theory: Green's functions and applications
This monograph is an excellent introduction to the mathematical techniques used to describe the scattering and propagation of scalar waves, in particular sound waves. The scalar wave equations and Green's functions are developed from fundamental principles and to the following main problems: plane wave and spherical wave from flat interfaces, and propagation in a two-layer liquid half-space (Pekeeris waveguide). The detailed discussion facilitates extension of the techniques to real situations.
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amplitude analytic assume asymptotic Bessel function boundary conditions branch cut branch point Chap complex consider constant continuous spectrum cos0 critical angle cylindrical wave decay defined delta function density derivative differential equation distortion eigenfunctions eigenvalues evaluate example expand exponential finite fluid follows Fourier transform frequency Green's function half plane Hankel function Helmholtz equation illustrated in Fig impedance incident integrand interface latter leaky poles line integral medium normal Note occur outgoing wave phase term plane wave point source positive propagation proper modes quadrant radial ray parameter real axis real virtual modes reflected field reflection coefficient residue result Retarded Green's Function saddle point satisfies scalar second sheet Sect Snell's law solution sound speed spatial spherical wave square root steepest descent steepest descent contour steepest descent path substitute surface tion transmitted field two-dimensional vanishes variable velocity potential wave equation waveguide wavenumber write written yields zero