Propagation of Ocean Waves Over a Shelf with Linear Transition

Abstract

A singular perturbation analysis based on the WKB (Wentzel–Kramers–Brillouin) technique to study the hydrodynamic performance of periodic ocean waves that are incident on a shelf with linear transition is proposed. We derive a linear model to predict the propagation of the long ocean waves over the shelf. In this manner, the spatial distribution for the surface elevation of the ocean waves over the shelf as a function of three dimensionless parameters, namely, a small kinematical parameter and two geometrical parameters, is governed by a second-order linear ordinary differential equation. The kinematical parameter denotes the ratio of the potential head, due to gravity, to the kinetic head of the ocean waves along the longitudinal axis of the parabolic channel. Meanwhile the geometrical parameters represent a characteristic depth ratio of the shelf and the shelf slope. Using matching conditions, simple expressions for the reflection and transmission coefficients are obtained.