Gouyon waves in water of finite depth

Propagation of periodic stationary weakly vortical gravitational waves on the free water surface is considered. Similar wave motion was studied by Gouyon (Ann de la Fac des Sci de l’Université de Toulouse Sér 4(22):1–55, 1958) in linear and quadratic approximations in small parameter of the wave’s steepness ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} for the deep water conditions. In this paper this result is considered for the water of finite depth. Contrary to Gouyon who used the Euler approach a study of wave’s motion is performed here basing on the method of the modified Lagrangian coordinates. The wave’s vorticity Ω is specified as a series in the small parameter of steepness ε in the form: Ω=∑n=1∞εn·Ωnb\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega =\sum_{n=1}^{\infty }{\varepsilon }^{n}\cdot {\Omega }_{n}\left(b\right)$$\end{document}, where Ωn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega }_{n}$$\end{document} are arbitrary functions of the vertical Lagrangian coordinate b. Explicit expressions for the coordinates of the liquid particle trajectories and pressure distribution are obtained for the first two orders of perturbation theory. The nonlinear proportional to ε correction to the wave velocity is determined.