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Waves over a sea floor of varying depth

In reality, the sea floor very rarely has a constant depth for any large distance. Indeed, most generally, the floor rises and falls. One could ask what effect this has on the surface waves. How does one model this situation? The naive method would be to simply apply the equations derived above, letting h vary as a function of x.

However, upon close observation of the sea one discovers that as depth decreases the waves slow down and the period remains constant. As a result, wavelengh decreases and amplitude increases. If the depth decreases too much the wave will get too tall or too steep and will break (as it does on a beach). A tsunami is an extreme example of this - it is a very fast-moving wave (with a small amplitude in deep water) that grows extremely large as it slows down at the coastline. It is quite clear that our models have no way of predicting this behavior.

Since our model must now account for a changing amplitude and changing wavelength, some new ideas must be included to our model. For instance, we must take into account the phase change as x varies. Recall that the wave number, k, has an effect on the phase of the oscillation. But, since k changes as x changes (k=2p/L), we must measure the cumulative effect on the phase change. In particular, we must integrate k to obtain the correct phase change.