Abstract
The solution of the wave equation is obtained for a fluid medium with a curvilinear velocity profile. The reciprocal of the velocity squared is represented by an nth-degree polynomial as a function of depth. This leads to depth-dependent special functions numerically evaluated from recurrence relations by digital subroutines. The general solution is expressed as an integral in the wave-number space. The integrand is analytic in the cut complex wavenumber plane except at a countably infinite number of poles that characterize the normal modes of propagation. The residue theorem transforms the integral expression to an infinite series, the normal modes contribution, plus a branch line integral that vanishes in the long-range limit. In the short-range limit, the branch line integral leads to a wave diminishing in amplitude as the inverse square of the range and having a phase velocity equal to the speed of sound in the ocean's bottom.
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