14/07/2024
368 LINEAR WAVE THEORY
That’s the theory most widely used to describe water waves. It gives a sinusoidal shape of the wave, results in the orbital particle motion and we can determine the wave length L and speed vc from given parameters., i.e. wave period T, water depth d and height H.
However, for the average person the maths behind LWT is fairly complex, Laplace-equation, boundary conditions, differential equations etc. So the question was, can we use simple hydraulic principles such as momentum and continuity to analyse water waves? And surprisingly, the answer is “yes”.
From observations of e.g. foam specks on the surface of the ocean, we can see that the foam (or the fluid particles) move a bit forward with the crest, and a bit backwards with the trough. This is shown nicely in a video with a buoy https://lsintspl3.wgbh.org/en-us/lesson/buac20-int-waterwavemotion/1 and, in a more abstract way, in the first picture. This means, that the particle remains on average at the same place. Since it also moves up and down, it performs a 2D-motion. It rotates about an average position in x and y, Assuming that its velocity is constant, this path of motion must be a circle. From this conclusion, we can easily see that the surface profile must be sinusoidal, second picture.
Now we can progress to the momentum balance. Here we assume that the particle velocity has a maximum at the wave crest and reduces linearly – since we know the (a) the effect of the wave reduces with depth and (b) that the particle velocity ab the bed must be zero. We can then draw write the momentum balance, third picture.
With a further assumption for deep water, namely that the wave amplitude ‘a’ is very small compared with the water depth d, i.e. a
