# Gravity-capillary waves in reduced models for wave-structure interactions

## TL;DR

This paper introduces a reduced model with a Sommerfeld-type boundary condition to accurately compute steady gravity-capillary waves in low-speed flows, addressing longstanding numerical challenges in wave-structure interaction simulations.

## Contribution

It presents a novel asymptotic reduction and boundary condition approach for steady gravity-capillary waves, enabling more accurate numerical solutions in low-speed regimes.

## Key findings

- Validated low-speed wave predictions using exponential asymptotics
- Demonstrated sensitivity of full equations to boundary condition errors
- Provided a practical method for resolving the radiation problem

## Abstract

In order to determine the steady-state subcritical gravity-capillary waves that are produced by potential flow past a wave-making body, it is typically necessary to impose a radiation condition that allows for capillary waves upstream, but disallows those corresponding to gravity. However, this radiation condition is not known a priori and consequently, the computation of accurate numerical solutions to the steady-state problem remains problematic. Although the physical model might be modified (e.g. with viscosity), recovery of the original problem is not always possible. In this work, we discuss the above radiation problem, and show how, in the low-speed limit, the steady gravity-capillary waves can be resolved using a Sommerfeld-type boundary condition applied to an asymptotically reduced set of water-wave equations. These results allow us to validate the specialized classes of low-speed waves theoretically predicted by Trinh & Chapman (2013) using methods in exponential asymptotics [J. Fluid Mech. 724, pp. 392--424]. The issues of numerically solving the full set of gravity-capillary equations for a potential flow are discussed, and the sensitivity to errors in the boundary conditions is clearly demonstrated.