Evolution of long water waves in variable channels

Abstract

This paper applies two theoretical wave models, namely the generalized channel Boussinesq (gcB) and the channel Korteweg-de Vries (cKdV) models (Teng & Wu 1992) to investigate the evolution, transmission and reflection of long water waves propagating in a convergent-divergent channel of arbitrary cross-section. A new simplified version of the gcB model is introduced based on neglecting the higher-order derivatives of channel variations. This simplification preserves the mass conservation property of the original gcB model, yet greatly facilitates applications and clarifies the effect of channel cross-section. A critical comparative study between the gcB and cKdV models is then pursued for predicting the evolution of long waves in variable channels. Regarding the integral properties, the gcB model is shown to conserve mass exactly whereas the cKdV model, being limited to unidirectional waves only, violates the mass conservation law by a significant margin and bears no waves which are reflected due to changes in channel cross-sectional area. Although theoretically both models imply adiabatic invariance for the wave energy, the gcB model exhibits numerically a greater accuracy than the cKdV model in conserving wave energy. In general, the gcB model is found to have excellent conservation properties and can be applied to predict both transmitted and reflected waves simultaneously. It also broadly agrees well with the experiments. A result of basic interest is that in spite of the weakness in conserving total mass and energy, the cKdV model is found to predict the transmitted waves in good agreement with the gcB model and with the experimental data available.

On the fission of algebraic solitons

“…The phase difference between two solutions decreases as the slope of a topography becomes smaller, since reflected waves neglected in the unidirectional model are insignificant in this case. A similar observation has been made for surface waves in comparison of numerical solutions between the uni-and bidirectional models (the KdV equation and the Boussinesq equations, respectively) by Teng & Wu (1993). To carry out numerical simulations beyond t = 300, a larger domain of computation is required for bidirectional waves, as indicated in figure 3a.…”

Boussinesq solution for solitary waves in uniform channels with sloping side walls

“…For nonrectangular channels whose width increases (decreases) from the bottom to the surface, is found to be greater (smaller) than one and, the further the channel geometry diers from a rectangle, the further the value diers from one. Detailed discussions on for dierent channel geometries can be found in references [2] to [4].…”

On the influence of wave reflection on shoaling and breaking solitary Waves

“…Several authors [15,19,21,28,30,40,43] have included the effect of smooth and slowly varying bottom topographies in both Boussinesq and shallow water theory. Wave shoaling is the effect by which surface waves propagating shorewards experience a decrease in the water depth.…”