Generalisations of water waves

Section 5.4 Generalisations of water waves

During the lectures and exercises, you will cover some of the following variants.

Linear water waves in finite depth: part of Exercise 5.5.2 and done in Problem Class Week 6.
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In lectures and the exercise, you will confirm that the dispersion relation in finite depth is modified to:

\begin{equation*} \omega = \omega_\text{finite} = \sqrt{gk \tanh(kh)}. \end{equation*}

In the limit $h \to \infty\text{,}$ since $\tanh(kh) \to 1\text{,}$ this approaches the deep-water dispersion relation of

\begin{equation*} \omega_\text{deep} = \sqrt{gk}, \end{equation*}

as shown in (5.3.14).

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Linear water waves in a current: part of Exercise 5.5.3 and done in Week 6 Thursday video.
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In these problems, you will modify the setup to include a horizontal current at infinity, with $\phi \sim Ux$ as $x \to -\infty\text{.}$ The main difference in this case is that there are additional terms to be included in the linear kinematic and dynamic conditions. As shown in the exercise, the dispersion relation is modified to

\begin{equation*} \omega = \omega_\text{current} = Uk \mp \sqrt{gk \tanh(kh)}. \end{equation*}

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There are now two distinct wavespeeds. In the version where there is zero current, $U = 0\text{,}$ the wavespeeds reduce to a single one (i.e. a single $|c|\text{,}$ with both left- and right-travelling waves; remember that we usually drop the $\pm$).
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Linear water waves in two fluids of different densities: part of Exercise 5.5.4 and done in Week 7 Problem Class.
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Consider that the interface $y - \eta$ separates two fluids of different densities, say $\rho_1\text{,}$ lying in the lower portion, $y < \eta\text{,}$ and $\rho_2$ in the upper portion, with $y > \eta\text{.}$ The main modification in the governing equations happens in Bernoulli’s equation (5.1.4).
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In this case, the linear Bernoulli equation on the surface is

\begin{equation} \rho_1\left( \pd{\phi_1}{t} + g\eta\right) = \rho_2\left(\pd{\phi_2}{t} + g\eta\right), \qquad \text{on $y = 0$}.\tag{5.4.1} \end{equation}

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The procedure is the same for this version, but the algebra is noticeably more involved. In this case, the dispersion relation is

\begin{equation*} \omega = \pm \sqrt{\left(\frac{\rho_1 - \rho_2}{\rho_1 + \rho_2}\right) gk}. \end{equation*}

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Linear water waves with surface tension: part of Exercise 5.5.5 and done in Week 7 Problem Class.
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For the case of deep water waves, this modifies the dispersion relation to

\begin{equation*} \omega = \sqrt{gk + \frac{Tk^3}{\rho}}. \end{equation*}

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