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THE FUNDAMENTAL EQUATIONS
PROPAGATION IN THE OCEAN AS A BOUNDARY VALUE
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acoustic pressure amplitude arbitrary assume becomes Bessel functions Bessel's equation boundary conditions branch cut Cartesian consider constant contour convergence coordinate system cylindrical defined density depth function derivative differential equation diffraction eigenfunctions eigenvalues energy expression fluid Fn(z frequency geometry given by Eq Green's function Hankel function Helmholtz equation homogeneous layer infinity inhomogeneous integral meters normal mode obtain oceanic waveguide orthonormal parameter particle phase velocities problem propagation radiation ray bundle ray theory approximation ray trace receiver depth refracted result rigid bottom sea surface Section separation of variables shadow zone shown in Figure smooth caustic solution of Eq solve sound speed sound velocity source depth spectrum speed of sound Sturm-Liouville Sturm-Liouville problem surface duct Theorem transmission loss turning point uniform convergence values vanish vector volume water column wave equation wave front wave number waveguide WKB approximation WKB solution Wronskian zero