Unsteady undular bores in fully nonlinear shallow-water theory
arXiv:nlin/0507029 · doi:10.1063/1.2175152
Abstract
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 wave front 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.
Revised version accepted for publication in Phys. Fluids, 51 pages, 9 figures
References in corpus (2)
Cited by in corpus (29)
- Dispersive shock waves and modulation theory
- Undular bore theory for the Gardner equation
- The theory of optical dispersive shock waves in photorefractive media
- Resonant radiation shed by dispersive shock waves
- Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations
- Dispersive Shock Waves in Viscously Deformable Media
- On the Galerkin / finite-element method for the Serre equations
- Extreme wave runup on a vertical cliff
- Shock waves in dispersive Eulerian fluids
- Asymptotic description of solitary wave trains in fully nonlinear shallow-water theory
- Dispersive shock waves theory for non-integrable equations
- Slow modulations of periodic waves in Hamiltonian PDEs, with application to capillary fluids
- Radiating dispersive shock waves in nonlocal optical media
- A Kinematic Conservation Law in Free Surface Flow
- On the nonlinear dynamics of the traveling-wave solutions of the Serre system
- Nonlinear Schrödinger equations and the universal description of dispersive shock wave structure
- Theory of quasi-simple dispersive shock waves and number of solitons evolved from a nonlinear pulse
- Dark solitons, dispersive shock waves, and transverse instabilities
- Evolution of wave pulses in fully nonlinear shallow-water theory
- Mechanical balance laws for fully nonlinear and weakly dispersive water waves
- Rankine--Hugoniot conditions for fluids whose energy depends on space and time derivatives of density
- Formation of dispersive shock waves in a saturable nonlinear medium
- Extended Lagrangian approach for the numerical study of multidimensional dispersive waves: applications to the Serre-Green-Naghdi equations
- Accessing and Manipulating Dispersive Shock Waves in a Nonlinear and Nonlocal Rydberg Medium
- Structure-preserving approximations of the Serre-Green-Naghdi equations in standard and hyperbolic form
- Numerical study of the Serre-Green-Naghdi equations and a fully dispersive counterpart
- Solitary wave-mean flow interaction in strongly nonlineardispersive shallow water waves
- Dispersive Dynamics in the Characteristic Moving Frame
- Evolution of intensive light pulses in a nonlinear medium with the Raman effect