ABSTRACT
Here, mainly long quadratic nonlinear waves are considered. Recall that although the term “long” waves is used here, the theory of these waves describes typical storm and swell waves in shallow seas, for example, in the North Sea. As shown in Section 1.4.4 (see also Figures 1.3 and 1.4), the equations used for this case are applicable up to values 2 (λ is the wavelength, h is the depth). So that you can study, for example, the “New Year” wave (8.6) described in Section 8.2 on their basis. However, nonlinear evolution of short waves is studied also. We use the theory presented in Section 1.5 for considering the last case.
It is reminded that these equations are written in Lagrangian coordinates. This approach opens a more simple way to study one-dimensional wave problems than the Euler’s method.
It is emphasized that the long waves can cover completely a submarine bank (underwater topography). In other words, the wave passes various zones of the resonant band. In this case, the elevation of the various points of the wave depends on the various points of the ssssresonant band. Therefore, different terms of governing equations play different roles in different points above the topography. As a result, shock waves, solitons, and elastica-like waves may be generated on the water above the topography. We emphasize that the latter result largely explains the occurrence of the Euler figures during the passage of underwater topographies.
