Initial surface disturbance on a shear current: the Cauchy--Poisson problem with a twist

TL;DR

This paper provides a comprehensive analytical solution to the classical Cauchy--Poisson problem for surface waves on a shear current, incorporating gravity, surface tension, and finite depth, revealing complex wave patterns influenced by shear.

Contribution

It presents the first general solution to the linear Cauchy--Poisson problem with shear, including effects of gravity, surface tension, and finite depth, and analyzes resulting wave patterns.

Findings

Strong shear causes diverse wave patterns and velocities.

Finite depth and capillarity weaken shear effects on wave patterns.

Shear modifies the dispersion relation, affecting wave behavior.

Abstract

We solve for the first time the classical linear Cauchy--Poisson problem for the time evolution an initial surface disturbance when a uniform shear current is present beneath the surface. The solution is general, including the effects of gravity, surface tension and constant finite depth. The particular case of an initially Gaussian disturbance of width $b$ is studied for different values of three system parameters: a "shear Froude number" $S\sqrt{b/g}$ ($S$ is the uniform vorticity), the Bond number and the depth relative to the initial perturbation width. Different phase and group velocity in different directions yield very different wave patterns in different parameter regimes when the shear is strong, and the well known pattern of diverging ring waves in the absence of shear can take on very different qualitative behaviours. For a given shear Froude number, both finite depth and nonzero capillary effects are found to weaken the influence of the shear on the resulting wave pattern. The various patterns are analysed and explained in light of the shear-modified dispersion relation.