Analytical Solution for Long-Wave Reflection by a General Breakwater or Trench with Curvilinear Slopes

Abstract

In the first part of this paper, an exact analytical solution in closed form for linear long-wave reflection by a submerged idealized breakwater or trench with various curvilinear slopes is given. The solution obtained finds almost all previous long-wave analytical solutions for wave reflection by idealized bathymetries to be its special cases, including the wave reflection by an infinite step, a continental shelf with a parabolic slope, a continental shelf with a linear slope, a rectangular obstacle, an obstacle of general trapezoidal shape with linear slopes, and a trench of general trapezoidal shape with linear slopes. In the second part, an exact analytical solution in the form of a Taylor series for linear long-wave reflection by a submerged quasi-idealized breakwater or trench is also constructed. It is shown by convergence analysis that the series solution converges in the entire physical domain. Based on the present analytical solutions, the reflection coefficients for long waves reflected by various breakwaters are calculated and the influence of the breakwater dimensions in the reflection effect is investigated. It is always found that the total reflection defined by the area under the reflection coefficient curve increases when the front and back slopes become steep. It is also found that the phenomenon of zero reflection for a symmetrical rectangular breakwater still remains for a general breakwater with curvilinear slopes as long as the bathymetry is symmetrical about the breakwater.