A note on impulsive sphere motion beneath a free-surface

A general framework is presented for solving the impulsive oblique motion of a spherical body in close proximity and below a free-surface. The fluid is considered to be impulsive and the flow as incompressible. The irrotational flow field is deduced from a velocity potential. The full nonlinear problem is reduced to a sequence of boundary-value problems by employing a small-time expansion technique. The mixed boundary conditions are of a Dirichlet type on the undisturbed free-surface and of a Neumann type on the equilibrium spherical shape. The solution is obtained by employing a Green's function and the method of multipoles expansions. General expressions, correct to each order in the small-time, are given for the free-surface deflections and the pressure force experienced by the moving sphere.