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THE FUNDAMENTAL EQUATIONS
APPROXIMATESOLUTIONS OF THE WAVE EQUATION
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acoustic pressure approximation arbitrary assume becomes Bessel functions Bessel's equation boundary conditions branch cut Cartesian coefficients complex consider constant contour convergence coordinate system cylindrical defined delta function density derivative differential equation duct eigenfunctions eigenvalues energy expression fluid frequency function f(z geometry given by Eq Green's function Hankel function Helmholtz equation homogeneous layer infinity inhomogeneous integral integrand interval meters obtain orthonormal parameter phase velocities propagation ray theory receiver depth refracted result rigid bottom saddle point satisfy the boundary sea surface Section separation of variables set of eigenfunctions shown in Figure side of Eq singularities smooth caustic ſº solution of Eq solve sound speed source depth Sturm-Liouville problem term theorem turning point uniform convergence values vanish vector volume wave equation wave front wave number waveguide WKB approximation WKB solution Y2di zero