Non-uniqueness in the problem of forward motion of bodies in a two-layer fluid

Abstract

The classical linear problem of ship waves, describing forward motion of rigid bodies with a constant speed in an unbounded heavy fluid having a free surface, is studied. A two-dimensional statement of the boundary value problem is considered in the case when the fluid consists of two layers of different density and the bodies are totally submerged in one of the layers. For an arbitrary geometry of bodies, it is known that the problem is uniquely solvable almost everywhere in the set of physically meaningful values of speed and a parameter characterizing stratification. In this paper, the existence of exceptional values is numerically confirmed by constructing examples of non-uniqueness. For this, the ideas suggested by Motygin & McIver (2009) for the case of homogeneous fluid are developed.