On some solitary and cnoidal wave diffraction solutions of the Green–Naghdi equations

Abstract

Nonlinear waves of the solitary and cnoidal types are studied in constant and variable water depths by use of the Irrotational Green–Naghdi (IGN) equations of different levels and the original Green–Naghdi (GN) equations (Level I). These equations, especially the IGN equations, have been established more recently than the classical water wave equations, therefore, only a handful applications of the equations are available. Moreover, their accuracies and the conditions under which they are applicable need to be studied. As a result, we consider a number of surface wave propagation and scattering problems that include soliton propagation and fission over a bump and onto a shelf, colliding solitons, soliton generation by an initial mound of water and diffraction of cnoidal waves due to a submerged bottom shelf, and compare the predictions with experimental data when available.