Analytical Modeling of the Wave Reflection by a Single Cycloidal Geotube or Trench Based on the Modified Mild-Slope Equation

Abstract

This study investigated the wave reflection and transmission caused by a single bar or trench with a cycloidal cross section. An exact solution was constructed, in terms of the Frobenius series, to the modified mild-slope equation (MMSE). The aforementioned solution was validated against a numerical solution based on the linear potential flow theory for a single bar or a single trench with maximum gradients below 1:1, and the two types of solutions agreed each other. The solution was then compared with a numerical solution to the MMSE for wave scattering by a parabolic bar, and the comparison showed that the two solutions were nearly identical when the two types of bars almost overlapped. However, a further comparison between the solution and a truncated Taylor series solution to the MMSE revealed that the latter was divergent because it overlooked a singularity in the MMSE. Finally, the influence of certain parameters of the bar or trench on the reflection coefficient were investigated. The peak value of the reflection coefficient was proportional to the relative bar height or trench depth, and the total reflection was proportional to the bar or trench steepness. The phase of the peak reflection coefficient increased (or shifted upward) with increasing submergence–width ratio.