Shallow Water Dynamics on Linear Shear Flows and Plane Beaches

Abstract

Long waves in shallow water propagating over a background shear flow towards a sloping beach are being investigated. The classical shallow-water equations are extended to incorporate both a background shear flow and a linear beach profile, resulting in a non-reducible hyperbolic system. Nevertheless, it is shown how several changes of variables based on the hodograph transform may be used to transform the system into a linear equation which may be solved exactly using the method of separation of variables. This method can be used to investigate the run-up of a long wave on a planar beach including the development of the shoreline.

References in corpus (5)

• Steady periodic water waves with constant vorticity: regularity and local bifurcation
• A nonlinear Schrödinger equation for water waves on finite depth with constant vorticity
• Long Wave Dynamics along a Convex Bottom
• Fully nonlinear long-waves models in presence of vorticity
• Traveling water waves with compactly supported vorticity

Cited by in corpus (2)

• Nonlinear water waves in shallow water in the presence of constant vorticity: A Whitham approach
• A consistent reduction of the two-layer shallow-water equations to an accurate one-layer spreading model