I. Run-Up of Ocean Waves on Beaches II. Nonlinear Waves in a Fluid-Filled Elastic Tube

Citation

Zhang, Jin E. (1996) I. Run-Up of Ocean Waves on Beaches II. Nonlinear Waves in a Fluid-Filled Elastic Tube. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/TGA4-F552. https://resolver.caltech.edu/CaltechETD:etd-01072008-105605

Abstract

Part I.

This study considers the three-dimensional run-up of long waves on a horizontally uniform beach of vertically constant or variable slope which is connected to an open ocean of uniform depth. An inviscid linear long-wave theory is first applied to obtain the fundamental solution for a uniform train of sinusoidal waves obliquely incident upon a uniform beach of variable downward slope without wave breaking. The linearly superposable solutions provide a basis for subsequent comparative studies when the nonlinear and dispersive effects are taken into account, both separately and jointly, thus providing a comprehensive prospect of the extents of influences due to these physical effects. These comparative results seem to be new.

By linear theory for waves at nearly grazing incidence, run-up is significant only for the waves in a set of eigenmodes being trapped within the beach at resonance with the exterior ocean waves. Fourier synthesis is employed to analyze a solitary wave and a train of cnoidal waves obliquely incident upon a sloping beach, with the nonlinear and dispersive effects neglected at this stage. Comparison is made between the present theory and the ray theory to ascertain a criterion of validity for the classical ray theory. 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. Currents of significant velocities are produced by waves at incidence angles about 45° and by grazing waves trapped on the beach. Also explored are the effects of the variable downward slope and curvature of a uniform beach on three-dimensional run-up and reflection of long waves.

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 scheme is shown to have a very high accuracy by comparison with some exact solutions of the shallow water equations. The maximum run-up of a solitary wave predicted by the shallow water equations depends on the initial location of the solitary wave and is not unique in value because the wave becomes increasingly more steepened given longer time to travel in the absence of the dispersive effects; it is in general larger than that predicted by the linear long-wave theory. The farther the initial solitary wave of the KdV form is imposed from the beach, the larger the maximum run-up it will reach.

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. The result for the gB model is slightly larger than the wave run-up predicted by linear long-wave theory. The dispersive effects tend to reduce the wave run-up either for linear system or for nonlinear system.

A three-dimensional process of wave run-up upon a vertical wall has also been studied.

Part II.

This part 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. The results of this study have extended the scope of this subject, which has been limited to dealing with unidirectional solitary waves only (Olsen and Shapiro 1967), by establishing an exact theory for bidirectional solitons of arbitrary shape. This class of solitons has several remarkable characteristics. These solitons may have arbitrary shape and arbitrary polarity (upward or downward), and all propagate with the same phase velocity. The last feature of wave velocity renders the interactions impossible between unidirectional waves. However, 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. An iteration scheme has been developed to integrate the system. The system is distinguished by the fact that any local initial disturbance released from a state of rest will become two solitons traveling to the opposite direction, and shocks do not form if initial value is continuous.

Thesis Files