Nonlinear transient gravity waves due to an initial free-surface elevation over a topography

The behavior of nonlinear, two-dimensional, transient gravity waves inside an incompressible, nonviscous and homogeneous fluid is predicted. These waves are the response of an initial free-surface elevation in an infinite channel with a general topography. A perturbation technique is used for the investigation. In contrast to the limitations made on the horizontal extent of the topography by the shallow-water theory, the present approach necessitates small vertical extent of the topography relative to its horizontal extent and to the mean waterʹs depth as well. The same limitation is assumed satisfied by the initial free-surface elevation. Solutions up to the second order are obtained, discussed and illustrated. The asymptotic behavior of the solutions is obtained and discussed. The expressions for the right going and the left going waves are separated. The case of a very shallow fluid is deduced and analyzed for such a model, which is free from the limitations assumed by the shallow water theory. The validity of the obtained solution is tested by comparing the lowest streamline with the bottom topography. The difference between the linear and the nonlinear theories, and the effect of the bottom topography on the resulting flow are illustrated.