We solve for the first time the classical linear Cauchy-Poisson problem—the time evolution of an initial surface disturbance—when a shear current of uniform vorticity 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” Sb/g (S is the 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 anal...

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

Flexural- and capillary-gravity waves due to fundamental singularities in an inviscid fluid of finite depth

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.

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

https://iopscience.iop.org/article/10.1016/j.fluiddyn.2005.12.001/pdf

Motion due to fundamental singularities in finite depth water with an elastic solid cover

Laplace transform technique is used to solve an initial value problem describing waves generated by a disturbance created at the surface of water covered by an inertial surface composed of a thin but uniform distribution of floating particles. Green's integral theorem produces the transformed potential function from which the form of the inertial surface is obtained as an infinite integral after taking Laplace inversion. The method of stationary phase is then employed to evaluate this integral approximately for large time and distance.

Water waves generated by disturbance at an inertial surface

On developpe une methode de transformee de Laplace pour resoudre les problemes de valeur initiale, le potentiel de vitesse decrivant la generation d'ondes de gravite capillaires infinitesimales dans un liquide au repos avec une surface inertielle composee de particules flottantes distribuees uniformement

/pdf/on-the-generation-of-water-waves-at-an-inertial-surface-gvnv8nzx2h.pdf

P. F. Rhodes-Robinson

Papers

On the generation of water waves at an inertial surface