Modelling Wave-by-Wave Transmission

Abstract

Climate change is raising the wave energy that reaches the coast. Sea level rise deepens the water near the shore, so larger and longer waves can travel further before they break, and warming oceans weaken coral reefs that once absorbed this energy. At the same time, coastal communities ask for protection that keeps the natural view of the sea. Submerged breakwaters meet both needs. They keep their crest below the water level, so they stay out of sight and let only part of the wave energy pass over the crest.

Design practice describes this transmission with a single bulk transmission coefficient 𝐾𝑡, the ratio of the transmitted to the incident significant wave height. This one number scales the whole sea state down at once. It says nothing about the individual waves that reach the water behind the structure, and nothing about the transmitted wave period. A structure behind the breakwater, such as a moored vessel or a revetment, responds to individual waves, and often to waves that are both high and long. Thus, a single coefficient hides the information that such a structure needs.

This study proposes a wave-by-wave model of the transmission over a submerged breakwater, built with a vine copula. The model is developed using the 32 small-scale physical model tests of van Gent et al. (2023). In each test, the incident and the transmitted individual waves are matched by a zero-crossing procedure, which pairs 30,924 of the 33,533 incident waves (92.2%). Each matched pair gives four normalized variables, the incident and transmitted wave heights and periods. The model is built in two parts. First, a marginal distribution is fitted to each of the four variables and its parameters are regressed on the bulk sea state. Second, a regular vine copula is selected to describe the dependence between the four variables, a single copula family is fitted at each of its edges, and the copula parameters are regressed on the bulk sea state as well. Both parts are driven by the same four bulk parameters of the sea state, so the joint distribution of the transmitted height and the transmitted period can be obtained for a new sea state without measuring its individual waves.

The predicted marginal distributions reconstruct the wave heights well, with 𝑅² = 0.99, and the wave periods less well. The selected vine is a D-vine, and it reproduces the measured four-dimensional dependence with 𝑅² = 0.99 on the rank scale. Compared against the single coefficient, the model shows that 𝐾𝑡 misallocates the transmission across the wave heights. It understates the transmission of the smallest waves, at 1.78 𝐾𝑡, and overstates that of the largest waves, at 0.83 𝐾𝑡, which break over the crest. Applied to every incident wave, the coefficient therefore over-predicts the largest transmitted waves, here the transmitted 𝐻2% by 9%. The model predicts the individual transmitted height more accurately than 𝐾𝑡 in all 32 tests, with a median 𝑅² of 0.57 against 0.38, and its 90% interval contains about 90% of the waves. It also predicts the transmitted period and the joint behaviour of the height and the period, which a single coefficient cannot. Note that the dependence between the transmitted height and the period makes a wave that is both high and long about 1.5 times as likely as it would be under independence. In a design example, accounting for this dependence doubles the estimated number of damaging waves.

It is concluded that a vine copula turns the single transmission coefficient into a full joint distribution of the transmitted wave height and period, and keeps the wave-to-wave variability and the dependence that a single coefficient discards. Thus, the model allows a probabilistic design of the water behind a submerged breakwater, and gives the transmitted period that current practice leaves out.