Unsteady undular bores in fully nonlinear shallow-water theory

G. A. El*, R. H. Grimshaw, N. F. Smyth

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

122 Citations (Scopus)
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We consider unsteady undular bores for a pair of coupled equations of Boussinesq-type which contain the familiar fully nonlinear dissipationless shallow-water dynamics and the leading-order fully nonlinear dispersive terms. This system contains one horizontal space dimension and time and can be systematically derived from the full Euler equations for irrotational flows with a free surface using a standard long-wave asymptotic expansion. In this context the system was first derived by Su and Gardner. It coincides with the one-dimensional flat-bottom reduction of the Green-Naghdi system and, additionally, has recently found a number of fluid dynamics applications other than the present context of shallow-water gravity waves. We then use the Whitham modulation theory for a one-phase periodic travelling wave to obtain an asymptotic analytical description of an undular bore in the Su-Gardner system for a full range of "depth" ratios across the bore. The positions of the leading and trailing edges of the undular bore and the amplitude of the leading solitary wave of the bore are found as functions of this "depth ratio." The formation of a partial undular bore with a rapidly varying finite-amplitude trailing wavefront is predicted for "depth ratios" across the bore exceeding 1.43. The analytical results from the modulation theory are shown to be in excellent agreement with full numerical solutions for the development of an undular bore in the Su-Gardner system.

Original languageEnglish
Article number027104
JournalPhysics of Fluids
Issue number2
Publication statusPublished - 22 Feb 2006
Externally publishedYes


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