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

## TL;DR

This paper introduces semi-explicit, highly accurate root-finding formulas for the water-wave dispersion relation, significantly improving the efficiency and robustness of the Hamiltonian coupled-mode theory in modeling fully nonlinear water waves over complex bathymetry.

## Contribution

The paper develops new semi-explicit root-finding formulas for the water-wave dispersion relation, enhancing the computational efficiency of the HCMT for nonlinear water-wave problems.

## Key findings

- Explicit formulas support rapid root calculation with high accuracy.
- Three iterations suffice for machine-precision roots in demanding cases.
- The approach improves the efficiency and robustness of nonlinear water-wave simulations.

## Abstract

The Hamiltonian coupled-mode theory (HCMT), recently derived by Athanassoulis and Papoutsellis [1], provides an efficient new approach for solving fully nonlinear water-wave problems over arbitrary bathymetry. In HCMT, heavy use is made of the roots of a local, water-wave dispersion relation with varying parameter, 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. 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 Householder's root finding method. Explicit approximate formulae of very good accuracy are obtained, which are adequate to support HCMT for many types of applications. In most demanding cases, e.g. very steep, deep-water waves, machine-accurate determination of the required roots is achieved by no more than three iterations, using the explicit forms as initial values. Exploiting this root-finding procedure in the HCMT, results in an efficient, numerical solver able to treat fully nonlinear water waves over arbitrary bathymetry. Applications to demanding nonlinear problems demonstrate the efficiency and the robustness of the present approach.