A guide for selecting periodic water wave theories - Le Méhauté (1976)’s graph revisited

This note provides guidelines for selecting appropriate analytical periodic water wave solutions for applications, based on two physical parameters, namely, frequency dispersion parameter (water depth divided by wavelength) and the nonlinearity parameter (wave height divided by wavelength). The guidelines are summarized in a graphic format in the two-parameters space. The new graph can be viewed as the quantification of the well-known Le Méhauté (1976)’s graph with quantitative demarcations between applicable analytical periodic water wave solutions. In the deep water and intermediate water regimes the fifth-order Stokes wave theories (Zhao and Liu, 2022) are employed for the construction of the graph, while in the shallower water regime, Fenton (1999)’s higher order cnoidal wave theories are used. The dividing lines between the applicable ranges of Stokes wave and cnoidal wave theories are determined by the values of Ursell number, a ratio between the nonlinearity and frequency dispersion.