Stability of Nearly Breaking Long Waves

by Nikolaos D. Katopodes, Univ of Michigan, Ann Arbor, United States,

Document Type: Proceeding Paper

Part of: Mechanics Computing in 1990's and Beyond


This paper addresses the development of equations for nonlinear dispersive waves that have improved stability properties over the commonly used Boussinesq-type equations. Boussinesq-type equations are unstable in the short-wave regime, which, although outside the range of applicability of the Boussinesq equations, do appear in the numerical solution of long wave problems as the latter approach the shoreline and begin to break. The basic hypothesis for the development of stable equations is that the Hamiltonian corresponding to Boussinesq-type equations either does not exist at all or becomes negative in the presence of very short waves. This is due to the insufficient approximation of the kinetic energy of the flow, which in the presence of short waves, fails to remain positive. The present work seeks approximations of the kinetic energy that remain positive regardless of wave length.

Subject Headings: Breaking waves | Boussinesq equations | Long waves | Nonlinear waves | Approximation methods | Water waves | Wave equations | Numerical methods

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