On the forward motion of an interface-crossing body in a two-layer fluid: the role of asymptotics in the problem's statement

A two-dimensional body moves forward with a constant velocity in an inviscid, incompressible fluid under gravity. The fluid consists of two layers having different densities, and the body is totally submerged so that it intersects the interface between the fluids. The resulting fluid motion is assumed to be steady-state in a coordinate system attached to the body. A well-posed statement of the problem for the velocity potentials is proposed in the framework of linearized water-wave theory. This statement consists of the so-called Neumann-Kelvin problem augmented by supplementary conditions, formulated in terms of the interface's elevation at the points, where the body intersects the interface. The analysis which leads to these two conditions involves the derivation of asymptotics near the intersection points and Green's representation for solutions to the Neumann-Kelvin problem. The uniqueness theorem for the suggested statement is proved under the assumption that the kinetic energy is finite. The asymptotics of a solution at infinity is found and an explicit formula for the resistance to forward motion is derived. The relationship of the formulae with the supplementary conditions is discussed.