I. Run-up of ocean waves on beaches. II. Nonlinear waves in a fluid-filled elastic tube

Part I is a study of three-dimensional run-up of long waves on a uniform beach of constant or variable slope which is connected to an open ocean of uniform depth. An inviscid linear long-wave theory is applied to obtain the solutions for a uniform train of sinusoidal, solitary and cnoidal waves obliquely incident upon a uniform beach without wave breaking. The wave-induced longshore current is evaluated by finding the Stokes drift of the fluid particles carried by the momentum of the waves obliquely incident upon a sloping beach. When the nonlinear effects are taken into account, the exact governing equations for determining a moving inviscid waterline are introduced here based on the local Lagrangian coordinates. A special numerical scheme has been developed for efficient evaluation of these governing equations. The maximum run-up of a solitary wave predicted by the shallow water equations depends on the initial location of the solitary wave. The dispersive effects are also very important in two-dimensional run-ups in its role of keeping the nonlinear effects balanced at equilibrium, so that the run-ups predicted by the generalized Boussinesq model (Wu 1979) always yield unique values for run-up of a given initial solitary wave, regardless of its initial position.; Part II is a study of nonlinear waves in a fluid-filled elastic tube, whose wall material satisfies the stress-strain law given by the kinetic theory of rubber. An exact theory for bidirectional solitons has been established. This class of solitons may have arbitrary shape and arbitrary polarity (upward or downward), and all propagate with the same phase velocity. The present new theory shows that bidirectional waves can have head-on collision through which our exact solution leaves each wave a specific phase shift as a permanent mark of the waves having made the nonlinear encounter. The system is at least tri-Hamiltonian and integrable.