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.
Waves with vorticity are common in nature (Constantin, 2005). However, most studies on water waves are devoted to irrotational flows (Stokes, 1847, 1880; Korteweg and de Vries, 1895; De, 1955; Skjelbreia and Hendrickson, 1960; Dean, 1965; Rienecker and Fenton, 1981; Fenton, 1988; Shin, 2016, 2018, 2019). The first rotational solution was described by Gerstner in 1802 and later independently rediscovered by Rankine (1863). A mathematical analysis of Gerstner’s solution was performed by Constantin (Henry, 2008). Kishida and Sobey (1988) calculated a third-order solution for waves on a constant shear current, and Dalrymple (1974) utilized a numerical method. Presumably, rotational solutions will be more important in the future (Fenton, 1990). Furthermore, Chen and Zou (2019) propounded that waves and currents coexist in the majority of marine environments, especially in the nearshore, estuarine, and coastal regions. Notably, wave-current interactions (Dean and Dalrymple, 1984; Boccotti, 2000) and vorticity (Nwogu, 2009) affect the dispersion relation. Whereas irrotational waves have been formulated from an Eulerian viewpoint, rotational waves have been formulated from Lagrangian (Constantin, 2005; Henry, 2008; Constantin et al., 2016) or Eulerian viewpoints (Dalrymple, 1974; Baddour and Song, 1988; Kishida and Sobey, 1988; Chen and Zou, 2019).