Solution for Water Waves with a Shear Current and Vorticity Over a Flat Bed

Rotational waves on shear currents have been extensively studied in the last several decades. Most previous studies focused on linear currents and constant vorticities, whose applications are limited. This study presents an analytical solution for water waves with a current and a nonvanishing vorticity that varies exponentially with depth over a flat bed. In contrast to previous studies, current and vorticity were primarily determined from the Euler equations. The solution satisfies the Euler equations, including all boundary conditions, and closely accords with well-known experimental data, which is in contrast to irrotational solutions. This study is valid for all water waves over a flat bed. Accordingly, it is possible to calculate steep waves near the breaking limit and ultra-shallow water waves near the solitary wave limit. Moreover, it was proved that the Euler equations are decomposed into Helmholtz equations for the velocity field and Bernoulli's equation for the pressure field. It was also proved that the direction of vorticity is the same as that of the particle motion and linear currents are inapplicable.