6.4: Linear Waves

We consider small amplitude waves in two dimensions \(x\) and \(z\), and call the surface deflection \(\eta(x, \, t)\), positive in the positive \(z\) direction. Within the fluid, conservation of mass holds, and on the surface we will assume that \(p = p_a\), the atmospheric pressure, which we can take to be zero since it is merely an offset to all the pressures under the surface. At the seafloor of course \(w = 0\) because flow cannot move into and out of the boundary; at the same time, because of the irrotational flow assumption, the velocity \(u\) can be nonzero right to the boundary.

Figure \(\PageIndex{1}\): the degrees of freedom available to moving water at the ocean surface vs. at the sea floor.

Taking a look at the relative size of terms, we see that \((u^2 + v^2 + w^2)/2\) is much smaller than \(gz\) - consider that waves have a frequency of one radian per second or so, with a characteristic (vertical) size scale of one meter, whereas \(g\) is of course order ten. Hence the Bernoulli equation near the surface simplifies to:

\[ \rho \dfrac{\partial \phi}{\partial t} + \rho g \eta \approx 0 \textrm{ at } z = 0. \]

Next, we note the simple fact from our definitions that

\[ \dfrac{\partial \eta}{\partial t} \approx \dfrac{\partial \phi}{\partial z} \textrm{ at } z = 0. \]

In words, the time rate of change of the surface elevation is the same as the \(z\)-derivative of the potential, namely \(w\). Combining these two equations we obtain

\[ \dfrac{\partial^2 \phi}{\partial t^2} + g \dfrac{\partial \phi}{\partial z} = 0 \textrm{ at } z = 0. \]

The solution for the surface is a traveling wave

\[ \eta(x, \, t) = a \cos (\omega t - kx + \psi), \]

where \(a\) is amplitude, \(\omega\) is the frequency, \(k\) is the wavenumber (see below), and \(\psi\) is a random phase angle. The traveling wave has speed \(\omega / k\). The corresponding candidate potential is

\begin{align} \phi (x, \, z, \, t) \, &= \, -\dfrac{a \omega}{k} \dfrac{\cosh (k(z+H))}{\sinh (kH)} \sin (\omega t - kx + \psi), \textrm{ where} \\[4pt] \omega \, &= \, 2 \pi / T \, = \, \sqrt{kg \tanh (kH)} \textrm{ (dispersion),} \\[4pt] k \, &= \, 2 \pi / \lambda. \end{align}

Here \(\lambda\) is the wavelength, the horizontal extent between crests. Let us confirm that this potential satisfies the requirements. First, does it solve Bernoulli’s equation at \(z = 0\)?

\begin{align} \dfrac{\partial^2 \phi}{\partial t^2} \, &= \, \dfrac{a \omega^3}{k} \dfrac{1}{\tanh kH} \sin (\omega t - kx + \psi) \\[4pt] &= \, a \omega g \sin (\omega t - kx + \psi) \textrm{ and} \\[4pt] \dfrac{\partial \phi}{\partial z} \, &= \, -a \omega \sin (\omega t - kx + \psi). \end{align}

Clearly Bernoulli’s equation at the surface is satisfied. Working with the various definitions, we have further

\begin{align} u(x, \, z, \, t) \, &= \, \dfrac{\partial \phi}{\partial x} \, = \, a \omega \dfrac{\cosh (k(z+H))}{\sinh (kH)} \cos (\omega t - kx + \psi), \\[4pt] w(x, \, z, \, t) \, & = \, \dfrac{\partial \phi}{\partial z} \, = \, -a \omega \dfrac{\sinh (k(z+H))}{\sinh (kH)} \sin (\omega t - kx + \psi), \\[4pt] p(x, \, z, \, t) \, &\approx \, -\rho \dfrac{\partial \phi}{\partial t} - \rho gz \\[4pt] &= \, \rho \dfrac{a \omega^2}{k} \dfrac{\cosh (k(z+H))}{\sinh (kH)} \cos (\omega t - kx + \psi) - \rho gz. \end{align}

At the surface \(z = 0\), it is clear that the hyperbolic sines in \(w(x, \, z, \, t)\) cancel. Then taking an integral on time easily recovers the expression given above for surface deflection \(\eta(x, \, t)\). The pressure here is \(\rho g \eta\), as would be expected. At depth \(z = -H\), \(w = 0\) because \(\sinh (0) = 0\), thus meeting the bottom boundary condition. The particle trajectories in the \(x\)-direction and the \(z\)-direction are respectively

\begin{align} \xi_p (x, \, z, \, t) \, &= \, a \dfrac{\cosh (k(z+H))}{\sinh (kH)} \sin (\omega t - kx + \psi) \\[4pt] \eta_p (x, \, z, \, t) \, &= \, \dfrac{a}{k} \dfrac{\cosh (k(z+H))}{\sinh (kH)} \cos (\omega t - kx + \psi). \end{align}

Hence the particles’ motions take the form of ellipses, moving clockwise when the wave is moving in the positive \(x\) direction.

Note that there are no nonlinear terms in \( [x, \, y, \, z, \, u, \, v, \, w, \, p, \, \phi] \) in any of these equations, and hence this model for waves is linear. In particular, this means that waves of different frequencies and phases can be superimposed, without changing the behavior of the independent waves.