Analytic Solution to the Modified Mild-Slope Equation for Reflection by a Rectangular Breakwater with Scour Trenches

Abstract

In this paper, an analytic solution to the modified mild-slope equation (MMSE) for wave reflection by a submerged rectangular breakwater with two scour trenches is explored. Because of the use of the MMSE with effects of the bottom curvature and the slope-squared terms, the solution is not only valid in the whole wave range from shallow water to deep water, but also valid for topographies not restricted to vary moderately. The present analytic solution includes an existing analytic long-wave solution as its special case, and the computing results show good agreement between two solutions, except for a slight difference when waves approach intermediate-wave range. It is found that this slight difference used to lead to an incorrect conclusion that the reflection coefficient for wave reflection by a rectangular breakwater or trench is a periodic function to the ratio of the breakwater length to the wavelength. This analysis shows that the reflection coefficient is a periodic oscillation function with a variable oscillation amplitude rather than a periodic function with a constant oscillation amplitude. It is also found that the discrepancy between the two solutions, respectively based on the MSE and the MMSE, mainly occurs for intermediate waves. Based on the present MMSE-based solution, the influence of trench dimensions on the reflection effect is investigated. It is shown that in the whole wave range, the phenomenon of zero reflection occurs more frequently for symmetrical bathymetry.