Semi-explicit solutions to the water-wave dispersion relation and their role in the non-linear Hamiltonian coupled-mode theory

Abstract

The Hamiltonian coupled-mode theory (HCMT), recently derived by Athanassoulis and Papoutsellis [Proceeding of 34th International Conference on Ocean Offshore Arctic Engineering, ASME, St. John’s, Newfoundland, Canada, 2015], provides an efficient new approach for solving fully non-linear water-wave problems over arbitrary bathymetry. This theory exactly transforms the free-boundary problem to a fixed-boundary one, with space and time-varying coefficients. In calculating these coefficients, heavy use is made of the roots of a local, water-wave dispersion relation with varying parameters, which have to be calculated at every horizontal position and every time instant. Thus, fast and accurate calculation of these roots, valid for all possible values of the varying parameter, are of fundamental importance for the efficient implementation of HCMT. In this paper, new, semi-explicit and highly accurate root-finding formulae are derived, especially for the roots corresponding to evanescent modes. The derivation is based on the successive application of a Picard-type iteration and the Householders root-finding method. Explicit approximate formulae of very good accuracy are obtained, and machine-accurate determination of the required roots is easily achieved by no more than three iterations, using the explicit forms as initial values. Exploiting this procedure in the HCMT, results in an efficient, dimensionally reduced, numerical solver able to treat fully non-linear water waves over arbitrary bathymetry. Applications to four demanding non-linear problems demonstrate the efficiency and the robustness of the present approach. Specifically, we consider the classical tests of strongly non-linear steady wave propagation and the transformation of regular waves due to trapezoidal and sinusoidal bathymetry. Novel results are also given for the disintegration of a solitary wave due to an abrupt deepening. The derived root-finding formulae can be used with any other multimodal methods as well.