2007/03/31 by G. A. El, G. A. EL, R. H. J. Grimshaw +3 · 1 citation
Earth and Planetary Sciences · Engineering · Physics and Astronomy · #Fluid Dynamics and Thin Films #Nonlinear Waves and Solitons #Ocean Waves and Remote Sensing #nlin.PS #nlin.SI
paper · pdf · doi:10.1017/s0022112007006817
published as J. Fluid Mech. 585, 213-244 (2007) · accepted for publication in J. Fluid Mech.
arxiv created 2007/03/31 · openalex publication_date 2007/08/07 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
This paper considers the propagation of shallow-water solitary and nonlinear periodic waves over a gradual slope with bottom friction in the framework of a variable-coefficient Korteweg–de Vries equation. We use the Whitham averaging method, using a recent development of this theory for perturbed integrable equations. This general approach enables us not only to improve known results on the adiabatic evolution of isolated solitary waves and periodic wave trains in the presence of variable topography and bottom friction, modelled by the Chezy law, but also, importantly, to study the effects of these factors on the propagation of undular bores, which are essentially unsteady in the system under consideration. In particular, it is shown that the combined action of variable topography and bottom friction generally imposes certain global restrictions on the undular bore propagation so that the evolution of the leading solitary wave can be substantially different from that of an isolated solitary wave with the same initial amplitude. This non-local effect is due to nonlinear wave interactions within the undular bore and can lead to an additional solitary wave amplitude growth, which cannot be predicted in the framework of the traditional adiabatic approach to the propagation of solitary waves in slowly varying media.