Analytical study for linear wave transformation by a trapezoidal breakwater or channel

In various field applications in coastal and ocean engineering such as design of offshore borrow pits, navigation channels, subaqueous dredged material placement sites and submerged breakwaters, numerical wave transformation models based on the mild-slope equation (MSE) are commonly used since trench/channel or shoal/breakwater slopes often do not exceed 1:1 in practice given these are controlled by the angle of repose of sediments or in the case of submerged breakwaters by design criteria which typically recommend slopes not exceed 1:1.5. MSE-based analytical solutions confirming their validity to represent important processes such as wave reflection are important. In this paper, an analytical solution to the modified MSE (MMSE) in terms of Taylor series is constructed for wave transformation by a trench or breakwater with general trapezoidal shape. This solution is superior to previous analytical solutions based on the long-wave equation and the MMSE. It is also superior to the step method and the eigenfunction expansion method based on the Laplace equation in the sense that approximation to sloped transitions by steps is avoided. It is shown through intensive comparisons and tests that the present analytical model is valid for a trench/channel or shoal/breakwater with the maximal slope being at least 1:1.5.

► Analytic solution to EMMSE is given for reflection by a trapezoidal trench or breakwater.
► Without limitation of wave condition, the solution is superior to all long wave solutions.
► This exact solution is clearly superior to MSE (MMSE)-based approximate analytic solutions.
► Without approximation to the two transitions, the solution is superior to the step method.
► h(x) is required to be in C2 here while in C∞C∞ in other MMSE-based exact solutions.