Subsection 5.3.1 Linearisation of the water-wave equations
Our task in this section is to linearise the water-wave equations given in
Theorem 5.1.2. The culprit to the difficulty in seeking solutions concerns two main issues. The first is the nonlinearity in Bernoulli’s equation (i.e. the term
\(|\nabla\phi|^2\)). The second is to do with the specification of the unknown free surface,
\(\eta(x, t)\text{,}\) and the need to solve the extra equations on this unknown location.
The idea of linearisation relies upon the consideration that if disturbances in the fluid are small, then the quadratic terms in the free-surface conditions can be neglected. One way to see this is to introduce a scaling on all the associated quantities. For example, we consider the free surface as written as
\begin{equation}
\eta(x, t) = \delta \overline{\eta}(x, t), \tag{5.3.1}
\end{equation}
where \(\delta \ll 1\) (\(\delta\) is a small number). The parameter \(\delta\) is simply a notational device to help remind us of the smallness of the free surface.
Now since we assume the motion in the fluid is also small, then we introduce a similar scaling on the potential function. We assume
\begin{equation}
\phi(x, y, t) = \delta \overline{\phi}(x, y, t), \tag{5.3.2}
\end{equation}
within the fluid. As a consequence, all velocities are small and of order \(\delta\text{.}\)
Now consider the evaluation of the vertical velocity on the surface. By Taylor’s formula, we can expand:
\begin{equation*}
v = \pd{\phi(x, \eta, t) }{y}= \pd{\phi(x, 0, t)}{y} + \pd{^2\phi(x, 0, t)}{y^2} \eta + \text{higher-order terms}.
\end{equation*}
In fact, if we use our assumptions
(5.3.1) and
(5.3.2), we can see the size of the vertical velocity more explicitly:
\begin{equation*}
v = \delta \pd{\overline{\phi}(x, 0, t)}{y} + \delta^2 \pd{^2\overline{\phi}(x, 0, t)}{y^2} \overline{\eta} + O(\delta^3).
\end{equation*}
We will essentially ignore all terms of \(\delta^2\) or smaller.
Let us consider now the kinematic condition
(5.1.6). Since the term with the horizontal velocity
\(u \pd{\eta}{x} = O(\delta^2)\text{,}\) then we see that the kinematic approximation now yields
\begin{equation*}
\delta\pd{\overline{\eta}}{t} = \delta \pd{\overline{\phi}}{y} + O(\delta^2), \qquad \text{on $y = 0$}.
\end{equation*}
As we indicated, the use of \(\delta\) is only to aid with ordering. We see that if we return back to the original variable scalings, then the above indicates that the approximate kinematic condition to be considered is:
\begin{equation*}
\pd{\eta}{t} \sim \pd{\phi}{y} \qquad \text{on $y = 0$}.
\end{equation*}
It is typical, when dealing with linear water waves, to dispense with the notation indicating "approximate", i.e. \(\sim\) here, on the understanding that the solutions are satisfied only approximately.
You will be able to practice this linearisation in the exercises.
The linearisation of Bernoulli’s equation in
(5.1.7) is done similarly, and the end result is that the term,
\begin{equation*}
|\nabla \phi|^2 = \left( \pd{\phi}{x}\right)^2 + \left( \pd{\phi}{y}\right)^2,
\end{equation*}
is assumed to be small compared to the other terms of the equation. Therefore, Bernoulli’s equation is then approximated as
\begin{equation*}
\pd{\eta}{t} + u \pd{\eta}{x} = v \qquad \text{on $y = \eta$}.
\end{equation*}
Theorem 5.3.2. Linear water wave equations.
Consider two-dimensional potential flow (inviscid, incompressible, irrotational) of fluid. A model for linear water waves is as follows:
\begin{align}
\nabla^2 \phi = 0 & \qquad \text{$-\infty < y < \eta$}\tag{5.3.3}\\
\pd{\phi}{t} + \frac{1}{2} |\nabla \phi|^2 + g\eta = 0 & \qquad \text{on $y = \eta$}\tag{5.3.4}\\
\pd{\eta}{t} + u \pd{\eta}{x} = v & \qquad \text{on $y = \eta$}\tag{5.3.5}
\end{align}
where the free surface is located at \(y = \eta(x, t)\text{.}\)
In water of infinite depth, we require that
\begin{equation}
\nabla\phi \to (0, 0) \qquad \text{as $y \to -\infty$},\tag{5.3.6}
\end{equation}
while for the case of finite depth, we have
\begin{equation}
\pd{\phi}{y} = 0 \qquad \text{on $y = -h$}.\tag{5.3.7}
\end{equation}
Subsection 5.3.2 Solving for linear waves in infinite depth
We now seek to solve the linear water wave equations in
Theorem 5.3.2.
We assume that the wave takes the form of a simple harmonic wave:
\begin{equation}
\eta(x, t) = A\cos(kx - \omega t). \tag{5.3.8}
\end{equation}
Examining the free-surface conditions in
(5.3.4) and
(5.3.5) suggests that the potential should take the form of a sinusoidal:
\begin{equation}
\phi(x, y, t) = f(y) \sin(kx - \omega t),\tag{5.3.9}
\end{equation}
which we shall verify is appropriate.
Firstly, substitution of
(5.3.9) into Laplace’s equation
(5.3.3) yields
\begin{equation}
f''(y) - k^2 f(y) = 0.\tag{5.3.10}
\end{equation}
Turning now to the two free-surface conditions
(5.3.4) and
(5.3.5), as well as the condition of infinite depth, we have the following necessary constraints:
\begin{align}
f'(0) \amp= \omega A, \tag{5.3.11}\\
\omega f(0) \amp= gA, \tag{5.3.12}\\
f(-\infty) \amp\to 0. \tag{5.3.13}
\end{align}
\begin{equation*}
f(y) = B \e^{ky} + D \e^{-ky},
\end{equation*}
and so the infinite depth condition
(5.3.6) yields
\(D \equiv 0\text{.}\) We substitute now
\(f(y) = B \e^{ky}\) into the two remaining conditions and simplify, yielding the matrix equation:
\begin{equation*}
\begin{pmatrix}
\omega \amp -k \\
g \amp -\omega
\end{pmatrix}
\begin{pmatrix}
A \\ B
\end{pmatrix}
=
\begin{pmatrix}
0 \\ 0
\end{pmatrix}.
\end{equation*}
Therefore, in order for there to be nontrivial solutions, the determinant of the matrix must ber zero, and this yields
\begin{equation*}
\omega^2 = gk,
\end{equation*}
or alternatively if we take the positive root,
\begin{equation}
\omega = \sqrt{gk}.\tag{5.3.14}
\end{equation}
The above expression is known as the
dispersion relation for water waves in deep water.
