Waves on shearing current for an arbitrary inertial viewer

TL;DR

This paper develops a generalized mathematical framework to analyze water waves propagating over shearing currents from the perspective of an arbitrary inertial observer, accounting for complex effects due to viewer motion.

Contribution

It introduces a new formulation of wave-current interactions that accounts for arbitrary inertial viewer motion, extending previous models limited to stationary or specific observer frames.

Findings

Derived the moving viewer dispersion relation for constant shear currents.

Showed that viewer motion affects wave behavior due to loss of Galilean invariance.

Provided a generalized boundary value problem formulation for shearing currents.

Abstract

Real world water waves often propagate on current. And, the measurement of waves and current is an important task for coastal and marine engineers. Modern marine measurement technologies (i.e. unmanned autonomous vehicles, drones) often propagate with different velocities relative to the current and waves. It has been shown that, due to a loss of Galilean invariance, viewer velocity has a non-trivial effect on the mathematical form of the wave and uniform current problem. It is demonstrated herein that similar complexities arise for shearing currents. The work provides a generalized formulation of the wave and shearing current boundary value problem for an arbitrary inertial viewer. The moving viewer dispersion relation for the special case of a constant current shear is also derived.