The analytic solution for hydrodynamic diffraction by submerged prolate spheroids in infinite water depth

This study investigates the linear hydrodynamic scattering problem by stationary prolate spheroidal bodies and aims at providing an analytic solution for the associated boundary value problem. It extends the work of the present author on the hydrodynamics of oblate spheroidal bodies following the same procedure. The structural model under consideration is a spheroid with its polar axis greater than its equatorial diameter, subjected to the action of monochromatic incident waves. The polar axis is assumed to be perpendicular to the free surface that leads to the axisymmetric case concept. The analytic solution is sought using the method of multipole expansions constructed by employing Thorne’s formulas (Multipole expansions in the theory of surface waves. Proc Cam Philos Soc 49:707–716, 1953) that describe the velocity potential at singular points within a fluid domain with free upper surface and infinite water depth. The final stage of the solution process is the application of the zero velocity condition on the wetted surface of the spheroid. Inevitably this task requires the transformation of the involved velocity potentials, originally expressed with respect to spherical and polar coordinates, into prolate spheroidal coordinates. To this end, the appropriate addition theorems are derived, which recast Thorne’s expressions into infinite series of associated Legendre functions.