Ocean Surface Waves

Figure 1.3: Wave on an ocean of depth $H$. The wave is moving to the right and the particles of water at the surface move up and down as shown by the small vertical arrows.

These waves are manifested as undulations of the ocean surface as seen in figure 1.3. The speed of ocean waves is given by the formula

$$c = \left( \frac{g \tanh (kH)}{k} \right)^{1/2} ,$$ (2.6)

where $g = 9.8 \mbox{ m} \mbox{ s}^{-2}$ is a constant related to the strength of the Earth's gravity, $H$ is the depth of the ocean, and the hyperbolic tangent is defined as^2.1

$$\tanh (x) = \frac{\exp (x) - \exp (-x)}{\exp (x) + \exp (-x)} .$$ (2.7)

Figure: Plot of the function $\tanh (x)$. The dashed line shows our approximation $\tanh (x) \approx x$ for $\vert x\vert \ll 1$.

As figure 1.4 shows, for $\vert x\vert \ll 1$, we can approximate the hyperbolic tangent by $\tanh (x) \approx x$, while for $\vert x\vert \gg 1$ it is $+1$ for $x > 0$ and $-1$ for $x < 0$. This leads to two limits: Since $x = k H$, the shallow water limit, which occurs when $kH \ll 1$, yields a wave speed of

$$c \approx (gH)^{1/2} ,~~~~ \mbox{(shallow water waves)} ,$$ (2.8)

while the deep water limit, which occurs when $kH \gg 1$, yields

$$c \approx (g/k)^{1/2} ,~~~~ \mbox{(deep water waves)} .$$ (2.9)

Notice that the speed of shallow water waves depends only on the depth of the water and on $g$. In other words, all shallow water waves move at the same speed. On the other hand, deep water waves of longer wavelength (and hence smaller wavenumber) move more rapidly than those with shorter wavelength. Waves for which the wave speed varies with wavelength are called dispersive. Thus, deep water waves are dispersive, while shallow water waves are non-dispersive.

For water waves with wavelengths of a few centimeters or less, surface tension becomes important to the dynamics of the waves. In the deep water case the wave speed at short wavelengths is actually given by the formula

$$c = (g/k + A k )^{1/2}$$ (2.10)

where the constant $A$ is related to an effect called surface tension. For an air-water interface near room temperature, $A \approx 74 \mbox{ cm}^3 \mbox{ s}^{-2}$.