Linear surface gravity waves on current for a general inertial viewer

Abstract

Marine measurement instrumentation, such as free-floating wave buoys, drones, and autonomous unmanned vehicles, often propagates in different directions and velocities relative to the fluid and waves. Convention assumes that these different instrumentations provide Galilean invariant descriptions of the wave field. Herein, it is shown that Galilean invariance exists for the water wave problem only in a restricted sense. The impact of this loss of invariance is investigated using a new formulation of the water wave problem, which is generalized for both current and an arbitrary inertial viewer. In the still water limit, the boundary value problem is shown to be non-invariant under Galilean transformations. This impacts the dispersion relation and interpretation of measurements. It also explains the appearance of wave modes on current, which have no analogy on still water. These modes do not appear in a still water formulation because it is a degenerate representation exhibiting a loss of Galilean symmetries. The approach provides a more complete solution of the wave–current boundary value problem by making a clear distinction between current and viewer velocity effects. Numerical examples that demonstrate the importance of the results on calculating wave characteristics are given.