Scattering of sound by a cylindrically symmetric seamount

A three-dimensional waveguide is used to model scattering of sound by a cylindrically symmetric seamount. The waveguide is "closed" in the sense that it has a boundary below the ocean bottom. This makes the spectrum of local wavenumbers discrete. Partial differential equations govern the modal expansion coefficients of the acoustic pressure and of the radial pressure gradient. Their solutions must be finite-valued at the axis of symmetry and satisfy outgoing radiation conditions at the perimeter of the base of the seamount. Direct numerical integration of this problem is unstable, but with variation of parameters and a Riccati transformation an equivalent problem is obtained for which numerical integration is stable. This approach is compared with an alternative [Pannatoni, POMA, Vol.14, 070003 (2011)] that uses leaky modes of an "open" waveguide having no boundary below the ocean bottom. It is shown how coupling between nearest-neighbor modes of the "closed" waveguide relates to the leaky modes of the "open" waveguide. This coupling restricts the step size that numerical integration can use. Some computations to validate this approach are displayed.