線形分散性と浅海長波の非線形性を合わせ持つモデル方程式
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概要
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The Korteweg-de Vries (KdV) and Boussinesq equations are representatives for shallow-water waves in dispersive systems. Both equations have soliton solutions and play significant roles in many nonlinear wave systems, such as in coastal engineering works and plasma dynamics (e.g., Laitone, 1960; Zabusky and Kruskal, 1965). In coastal engineering, the cnoidal wave (Laitone, 1960), the periodic solution of the KdV equation, is employed to describe properties of shallow-water waves. The Boussinesq-type equation (Peregrine, 1966, 1967), however, is on the recent trend to be used. The main reason is in the difference between their dispersion relations. Though both dispersion relations are the same for long wave approximation, for shorter period waves the dispersion relation of the KdV equation is unbounded, whereas that of the Boussinesq-type equation takes finite values. The Boussinesq-type equation, therefore, has possibility of applying in numerical calculation not only to the original long wave field but to the shorter period wave field. However, these two equations cannot show the cusp that is the limiting form of a wave when breaks on a gentle slope. Whitham (1967) and Benjamin (1967) independently proposed, on this matter, a model equation combining long wave nonlinearity with linear dispersion, for taking account of the effects from shorter period waves. The integro-differential equation suggested by Whitham and Benjamin is effective to describe waves propagating to the specified direction, and then this fact is a limitation in application of the equation. In addition, it is difticult to handle the equation in numerical works because of singularity in the kernel. The present paper, therefore, develops the concept of Whitham to the equations of Peregrine (1967) and offers a model of equations for long waves in three-dimensional, nonlinear dispersive systems, to be used in prediction of wave deformation in the coastal zone.
- 琉球大学の論文
- 1994-09-01
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