Question

Basic equtions in the linearised theory of water waves

Answer 1

Looking out to sea from the shore, we can see waves on the sea surface. Looking carefully, we notice the waves are undulations of the sea surface with a height of around a meter, where height is the vertical distance between the bottom of a trough and the top of a nearby crest. The wavelength, which we might take to be the distance between prominent crests, is around 50m - 100m. Watching the waves for a few minutes, we notice that wave-height and wave-length are not constant. The heights vary randomly in time and space, and the statistical properties of the waves, such as the mean height averaged for a few hundred waves, change from day to day. These prominent offshore waves are generated by wind. Sometimes the local wind generates the waves, other times distant storms generate waves which ultimately reach the coast. For example, waves breaking on the Southern California coast on a summer day may come from vast storms offshore of Antarctica 10,000km away.

If we watch closely for a long time, we notice that sea level changes from hour to hour. Over a period of a day, sea level increases and decreases relative to a point on the shore by about a meter. The slow rise and fall of sea level is due to the tides, another type of wave on the sea surface. Tides have wavelengths of thousands of kilometers, and they are generated by the slow, very small changes in gravity due to the motion of the sun and the moon relative to Earth.

Surface waves are inherently nonlinear: The solution of the equations of motion depends on the surface boundary conditions, but the surface boundary conditions are the waves we wish to calculate. How can we proceed?

We begin by assuming that the amplitude of waves on the water surface is infinitely small so the surface is almost exactly a plane. To simplify the mathematics, we can also assume that the flow is 2-dimensional with waves traveling in the x-direction. We also assume that the Coriolis force and viscosity can be neglected. If we retain rotation, we get Kelvin waves.

With these assumptions, the sea-surface elevation [math] \zeta\ [/math] of a wave traveling in the [math]x[/math]direction is:

[math] \zeta\ = a \sin(kx - \omega t) \,\![/math]

with

[math] \omega = 2 \pi\,f = \frac{2\pi}{T}; \qquad k = \frac{2\pi}{\lambda} \,\![/math]

where [math]\omega[/math] is wave frequency in radians per second, [math]f[/math] is the wave frequency in Hertz (Hz), [math]k[/math] is wave number, [math]T[/math] is wave period, [math]\lambda[/math] is wave-length, and where we assume, as stated above, that [math]k a = O (0)[/math].

The wave period [math]T[/math] is the time it takes two successive wave crests or troughs to pass a fixed point. The wave-length [math]\lambda[/math] is the distance between two successive wave crests or troughs at a fixed time.