Surface layers and linearized water waves: a boundary integral equation framework

TL;DR

This paper develops a boundary integral equation framework for modeling linearized surface water waves affected by floating plates or membranes with holes, providing new algorithms for efficient and scalable solutions.

Contribution

It introduces a general integral equation approach for complex boundary conditions involving plates and membranes with holes, analyzing their properties and developing fast numerical algorithms.

Findings

Boundary integral equations are Fredholm second kind.

Algorithms demonstrate robustness and scalability.

Numerical examples validate the approach.

Abstract

The dynamics of surface waves traveling along the boundary of a liquid medium are changed by the presence of floating plates and membranes, contributing to a number of important phenomena in a wide range of applications. Mathematically, if the fluid is only partly covered by a plate or membrane, the order of derivatives of the surface-boundary conditions jump between regions of the surface. In this work, we consider a general class of problems for infinite depth linearized surface waves in which the plate or membrane has a compact hole or multiple holes. For this class of problems, we describe a general integral equation approach, and for two important examples, the partial membrane and the polynya, we analyze the resulting boundary integral equations. In particular, we show that they are Fredholm second kind and discuss key properties of their solutions. We develop flexible and fast algorithms for discretizing and solving these equations, and demonstrate their robustness and scalability in resolving surface wave phenomena through several numerical examples.