Shoaling waves : Numerical solution of exact equations
Recently Longuet-Higgins and Cokelet have developed a method for the study of time-dependent two-dimensional irrotational free surface problems, which they applied to spatially-periodic waves in water of infinite depth.
Their approach may be extended to the generation and propagation of disturbances over a bed of arbitrary shape and motion. Kinematic and dynamic equations at the free surface and the bed may be expressed as Lagrangian differential equations for the motion of particles in terms of velocity potential and its derivatives. These are obtained from the numerical solution of a mixed boundary value problem formulated as an integral equation. For any free surface configuration and velocity potential distribution, the integral equation may be solved to give the velocities on the bed and surface, which may be integrated to give the subsequent form of the surface, and the process repeated.
Large amplitude motions on boundaries of irregular shape may be studied, as the method is analytically exact, provided the initial conditions are known.In the present study we examine the motion of disturbances as they approach a coastline. The method may be applied to many two-dimensional problems where magnitude of disturbance and time-dependence have rendered solution previously difficult.
KeywordsIntegral Equation Free Surface Solitary Wave Solid Boundary Finite Depth
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