Water Waves Generated by a Slowly Moving Two-Dimensional Body, Part I

Abstract

Low-speed motion of a ship leads to a singular perturbation problem as speed approaches zero, since waves occur only in a boundary layer of vanishing thickness at the free surface. The complete solution consists of the singular expansion superposed on a regular expansion. The latter (the "naive expansion") by itself satisfies all conditions in the lower half_space below the undisturbed free surface, but it does not represent the boundary layer. For the case of a two-dimensional surface_piercing body, it is shown that the regular perturbation series fails to satisfy the body boundary condition in a small wetted region just above the level of the undisturbed free surface. This fact leads to a nonhomogeneous body boundary condition that must be satisfied by the singular expansion. Without Such a condition, the singular part of the solution (which represents the real wave motion) would satisfy purely homogeneous conditions and thus would be indeterminate.