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

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.

27 pages, 11 figures, Accepted for Physics of Fluids

References in corpus (4)

• Kelvin wake pattern at large Froude numbers
• Ship waves in the presence of uniform vorticity
• How linear surface waves are affected by a current with constant vorticity
• Capillary-Gravity Waves on Depth-Dependent Currents: Consequences for the Wave Resistance

Cited by in corpus (14)

• Approximate dispersion relations for waves on arbitrary shear flows
• Ship waves on uniform shear current at finite depth: wave resistance and critical velocity
• A framework for modelling linear surface waves on shear currents in slowly varying waters
• Oblique waves on a vertically sheared current are rotational
• Transient wave resistance upon a real shear current
• Surface waves on arbitrary vertically-sheared currents
• Long ring waves in a stratified fluid over a shear flow
• Observation of surface wave patterns modified by sub-surface shear currents
• Nonlinear ring waves in a two-layer fluid
• Weakly non-linear transient waves on a shear current: Ring waves and skewed Langmuir rolls
• Waves from an oscillating point source with a free surface in the presence of a shear current
• Multiple resonances of a moving, oscillating surface disturbance on a shear current
• Wavefronts and modal structure of long surface and internal ring waves on a parallel shear current
• Long internal ring waves in a two-layer fluid with an upper-layer current