Numerical computations for A zero transmission of an incident wave in a three dimensional channel

Abstract

Numerical computations are made for the reflection and transmission of water waves propagating in a sidestepped channel and a channel with a side trench (or a series of side trenches). The physical model is to find the characteristics of the nearly zero transmission of the waves past a side trench (or trenches) for a possible application to a calm channel past a side trench in a pier. A mathematical model is formulated in the scope of the potential theory for a three-dimensional channel with various shapes of the cross sections. We developed two numerical methods for the problem. One is based on an analytical method, i.e., the application of the eigen-function expansions with an appropriate matching at the interfaces between two adjacent fluid sub-domains and the other is the localized finite element method based on a variational principle which can handle a more general geometry of the fluid domain. Initially we have planned to develop the present two methods as well as a series of experiments in parallel with the computations. But, the experimental measurements are not available yet. Therefore only the results of two different theoretical methods are presented herein. Comparisons between analytic solutions and LFEM results show a good agreement. Through systematic computations for various shapes of the side trenches, we present the frequencies of the incident waves which reflect nearly totally along the channel with a specific side trench. In the present study, it is found that the total reflection occurs at roughly two different modes of wave motions, i.e., one is the resonance mode of wave motions across the cross-section at the side trench, which is perpendicular to the direction of the incident wave propagation. The other is the waves trapped longitudinally in the trench which is in parallel with the incident waves.