Properties of the Energy Transport for Plane-Parallel Polychromatic Surface Gravity Waves in Waters of Arbitrary Depth

It is well known that the energy transport of ocean waves propagates with the group velocity and that the energy decreases exponentially with depth. Expanding this theory, we will derive expressions for the energy transport as a function of depth and the total instantaneous transport´s development over time for waves in waters of finite depth. Solutions to the Laplace equation are found for plane-parallel polychromatic waves with linearized boundary conditions. A time series of wave elevation collected at Uppsala University´s wave energy research test site is chosen to present the results. Solutions for waters of both infinite and arbitrary depths are presented and compared. The solutions are convolution-type integrals with the wave elevation where we have found efficient ways to calculate the kernels. The difference in group velocity between finite depth and infinite depth and its impact on the energy transport is clearly seen in the results. The use of the deep-water approximation gives a too low energy transport in the time averaged as well as in the total instantaneous energy transport. We further show that the total instantaneous energy transport can actually have a direction that is opposite to the direction of the waves as observed from a reference frame fixed to the seabed.