In this study, we present a solution for water waves with shear current and non-vanishing vorticity over a flat bed. The solution satisfies all boundary conditions and governing equations including the Euler equation. It is shown that two types of vorticities are possible according to their direction. The results accord closely with well-known experimental data rather than irrotational solution.


When the author was preparing the study (Shin, 2018, 2019), it was observed that there are small differences between the irrotational solution and experimental results of Le Méhauté et al (1968). This study is prepared to prove that the differences are resulted by the vorticity and shear current, i.e., a rotational solution is presented in this study.

While most studies of water waves are devoted to irrotational flows (Stokes, 1847, 1880; De, 1955; Skjelbreia et al., 1960; Korteweg et al., 1895; Dean, 1965; Rienecker et al., 1981; Fenton 1988, Shin, 2016, 2018, 2019), waves with vorticity are commonly seen in nature (Constantin, 2005). The first rotational solution was described by Gerstner in 1802 and was independently re-discovered later by Rankine (1863). A mathematical analysis of Gerstner's solution was performed by Constantin (Henry, 2008). For waves on a constant shear flow, Kishida et al (1988), calculated a third-order solution and Dalymple (1974), used a numerical method. As indicated by Fenton (1990), it is conceivable that in future rotational solutions will be more important. Furthermore, as indicated by Chen (2019), waves and currents coexist in the majority of marine environments, especially in the nearshore, estuarine and coastal regions. Wave-current interactions affect the dispersion relation (Dean, 1991; Boccotti, 2000) and vorticity also affect it (Nwogu, 2009).

While irrotational waves have been formulated with the Eulerian viewpoint, rotational waves have been formulated with the Lagrangian viewpoint (Constantin, 2005; Henry, 2008) or Eulerian viewpoint (Kishida, 1988; Dalrymple, 1975; Chen et al. 2019). The Eulerian viewpoint and Fourier expansions are adopted in this study.

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