Surface waves
Contents
Surface waves¶
We will open with waves on the surface of an inviscid fluid, bounded from above by another fluid (think the ocean and atmosphere), in a small-amplitude regime so that we can linearize the equations of motion about a static configuration with a flat interface. As with any physical small oscillation, waves arise from a restoring force. For surface waves the sources of a restoring force are gravity and surface tension. Waves whose dynamics are dominated by gravity are called surface gravity waves, while those with dynamics dominated by surface tension are called capillary waves.
(These are distinct from gravitational waves of the kind measured by LIGO!).
Dispersion relations¶
Before launching into a detailed mathematical analysis, we can already estimate the very different dispersion relations for each of these waves, in the “deep water” limit that the depth of the water is so large that it does not impact surface waves.
Surface gravity waves¶
First let us consider waves for which the restoring force is gravity. Absent viscosity, we wish to relate \(\omega\), with units of \(1/time\), to \(k\) with units of \(1/length\). The only additional consant availabe is \(g\) with units of \(length/(time)^2\), leading to
with \(c\) a constant (it is in fact 1).
Note that the phase velocity is \(v_p = \sqrt{\frac{g}{k}}\) and the group velocity \(v_g = \frac{1}{2} \sqrt{\frac{g}{k}\). The upshow is that long wavelength (low \(k\)) modes travel more quickly.
Capillary waves¶
Next consider the case that surface tension is the dominant restoring force. The surface tension is \(\alpha \sim (mass)/(time)^2 = (force)/(length)\). To relate \(\omega\) and \(k\) we need another quantity with mass dimension in it ;the density is the obvopus answer. So \(\alpha/\rho \sim (length)^3/(time)^2\). thus, we have
In this case \(v_p = \alpha \sqrt{k}/\rho\) and \(v_g = \frac{3\alpha \sqrt{k}}{2\rho}\); so long wavelengths (smaall \(k\)) travel more slowly than short wavelength ones.