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

Travis Askham, Tristan Goodwill, Jeremy G Hoskins, Peter Nekrasov, Manas Rachh

TL;DR

The paper tackles the problem of linear surface waves on deep water when the surface boundary contains holes in a plate or membrane, producing mixed-order boundary conditions across regions.It develops a nested boundary-integral representation by writing the velocity potential as a 3D Laplace single-layer $S_{3d}$ acting on a surface density $mu$ and using a Green's function $G_S$ to reduce the exterior problem to a 2D problem for $mu$ and $eta$, yielding a Fredholm second-kind system on $\Omega$.The authors prove invertibility of the resulting integral equations under a dissipative regime and demonstrate scalable numerics via a precorrected FFT to achieve $O(h^{-2}\log(h^{-1}))$ per iteration, with applications to capillary-gravity and flexural-gravity phenomena including polynyas and ice-rift geometries.Overall, the framework enables accurate, efficient modeling of complex surface-wave interactions in geophysical and engineering contexts where mixed-order boundary conditions arise.

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.

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

TL;DR

The paper tackles the problem of linear surface waves on deep water when the surface boundary contains holes in a plate or membrane, producing mixed-order boundary conditions across regions.It develops a nested boundary-integral representation by writing the velocity potential as a 3D Laplace single-layer $S_{3d}$ acting on a surface density $mu$ and using a Green's function $G_S$ to reduce the exterior problem to a 2D problem for $mu$ and $eta$, yielding a Fredholm second-kind system on $\Omega$.The authors prove invertibility of the resulting integral equations under a dissipative regime and demonstrate scalable numerics via a precorrected FFT to achieve $O(h^{-2}\log(h^{-1}))$ per iteration, with applications to capillary-gravity and flexural-gravity phenomena including polynyas and ice-rift geometries.Overall, the framework enables accurate, efficient modeling of complex surface-wave interactions in geophysical and engineering contexts where mixed-order boundary conditions arise.

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.
Paper Structure (13 sections, 17 theorems, 74 equations, 7 figures, 1 table)

This paper contains 13 sections, 17 theorems, 74 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Overview of the method
  3. Notation and assumptions
  4. Green's functions and regularity of integral operators
  5. The integral representations
  6. Exterior capillary-gravity waves
  7. Exterior flexural-gravity waves
  8. Numerical implementation and scalability
  9. Discretization of integral operators
  10. Scaling to large examples
  11. Examples and applications
  12. Discussion and future work
  13. Acknowledgements

Key Result

Proposition 3.2

Suppose that $p(z) = a_{d_p} z^{d_p} + \cdots + a_1 z + a_0$ with $d_p\geq 1$. If $a_1,\ldots,a_{d_p-1} \geq 0$, $a_{d_p}>0$, and $a_0$ is real, then $P(z) = (zp(z^2)-1)/2$ has precisely one positive real root, which we take to be $\rho_1$. In this case, $G_{\operatorname{S}}({\mathbf{r}})$ and $G_\ and likewise for their derivatives, as $|{\mathbf{r}}|\to\infty \; .$ In the case that the coeffici

Figures (7)

  • Figure 1: General setup of the problem.
  • Figure 1: Discretization of a geometry (left) and convergence of surface-volume operators (right). The geometry uses 8th-order Vioreanu-Rokhlin nodes inside and 16th-order Gauss-Legendre panels on the boundary. The integral operators were applied to a smooth density on a disk and compared to a reference value for a point in the domain. The dashed line represents 8th-order convergence.
  • Figure 1: Exterior capillary-gravity waves with Neumann boundary conditions. The real part of $\partial_z \phi$ is plotted on the left, while the $\log_{10}$ relative self-convergence error is plotted on the right.
  • Figure 2: The geometry for the spreading algorithm. A given source (black dot) is spread to the nearest $n\times n$ subset of the equispaced grid (pink stars). The equivalent charges on the $n\times n$ grid are chosen so that the fields agree on a series of proxy rings (salmon rings).
  • Figure 2: Exterior flexural-gravity waves with the free plate boundary conditions. The real part of $\partial_z \phi$ is plotted on the left, while the $\log_{10}$ relative error of self-convergence is plotted on the right.
  • ...and 2 more figures

Theorems & Definitions (32)

  • Remark 3.1
  • Proposition 3.2
  • Proof 1
  • Definition 3.3: Dissipative regime
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Proof 2
  • Theorem 3.7: Trace theorem
  • Lemma 3.8: Rellich's lemma
  • ...and 22 more