A Weak Formulation of Water Waves in Surface Variables

In 2006, Ablowitz, Fokas, and Musslamini published a paper where they reformulated the water-wave problem for irrotational fluids as a system of equations in surface variables given by

[mathjax]\begin{align*}
\displaystyle 0&= q_t + \frac{1}{2}q_x^2 + g\eta -\frac{1}{2}\frac{(\eta_t+q_x\eta_x)^2}{1 + \eta_x^2}\\
\displaystyle 0&= \int_{-\infty}^{\infty} e^{-ikx}\left(\eta_t\cosh(k(\eta+h)) + iq_x\sinh(k(\eta+h))\right)\:dx, \qquad \forall k \in \mathbb{R}.\end{align*}

In this formulation, \(q(x,t)\) is the trace of the velocity potential along the free surface \(z = \eta(x,t)\).

In A Weak Formulation of Water Waves in Surface Variables (short note posted on arXiv.org), I take a look at extending their formulation to a coupled system of integro-differential equations given by

\begin{align*}
0&=\int_{-\infty}^{\infty} e^{-ikx}\left(\left(q_x\eta_t – \eta_x(q_t+g\eta)\right)\cosh(k(\eta+h)) -i\left(q_t + g\eta\right)\sinh(k(\eta+h))\right) dx\\
0&=\int_{-\infty}^{\infty} e^{-ikx}\left(\eta_t\cosh(k(\eta+h)) + iq_x\sinh(k(\eta+h))\right)\:dx, \qquad \forall k \in \mathbb{R}.\end{align*}

You might be asking, why?

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Updates – in brief

It has been a while.  I’ve been working on quite a few projects with a few students.  I’ve made significant progress on the conservation laws project as well as the time-dependent pressure problem. 

Conservation Laws Project

I recently gave a talk on this in the “Waves in One World” Series.  You can find a direct link to the talk as well as the corrected  slides below.  I’ll write more information about this in an upcoming post once we have completed the pre-prints.  

Talk Abstract: We consider a nonlocal formulation of the water-wave problem for a free surface with an irrotational flow, and show how the problem can be reduced to a singleequation for the interface. The formulation is also extended to constant vorticity and interfacial flows of different density fluids. We show how this formulationcan be used to systematically derive Olver’s conservation laws not only for an irrotational fluid, but for constant vorticity and interfaces. This framework easily lends itself to computing the related conservation laws for various asymptotic models.

Waves in One World – Online Seminar – April 22, 2020
Recording of Talk
Annotated Slides (.pdf)

Time-Dependent Pressure Problem

Along the way with the conservation law problem, we discovered a direct map between the pressure at the bottom and the free-surface variables. I’ll detail more about this soon.

AIMS 2018: Taipei (Day 5)

There was an interesting talk on a wave-energy extraction device given by Tian-Shiang Yang (National Cheng Kung University, Taiwan).The question considered here seeks to determine how to relate the pressure prescribed on the free surface with the movement of the side pinned walls.  These walls will of course be coupled to a mass-spring-damper system.

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