From Sect. 3.4 we know that we may think of an irregular wave train as a sum of sine waves with various periods. For not too long periods of time (maybe an hour and distances of tens of kilometres) and for small amplitudes these sine waves have constant amplitudes and random phases. They travel in many different directions, all at their own velocity given by the so-called dispersion relation according to Airy or linear wave theory.
Wave fields disperse (spread out) since the different harmonic components travel at different speeds that depend on their frequency. In Sect. 3.5.2 we referred to this phenomenon as frequency dispersion. From the dispersion relation it becomes clear that longer waves travel faster than shorter waves. Also, we have seen that the group velocity (the velocity of the front) is larger for longer period waves.
At some distance from the storm centre one would therefore first experience a long fast travelling swell and later an increasingly shorter wave period. At long distances from the storm centre the shorter waves are filtered out since dissipation processes (due to currents, white-capping) more strongly affect the shorter waves3. As a result only a long, and fairly regular (as the various components travel at different speeds) swell remains. Besides, the swell is uni-directional crested because only waves travelling in a particular direction end up at a certain location away from the storm centre. The spreading due to different directions of propagation is called direction dispersion. Due to frequency and direction dispersion the spectrum of swell is narrow in frequency and direction respectively. As a result of spreading (and energy dissipation) swell is relatively low. Figure 3.12 shows swell waves arriving at the coast of Angola that have been generated in two different storms.
The characteristics of swell waves at a particular (coastal) location are determined by the characteristics of the storm and the distance to the storm. Swell can travel the oceans for thousands of kilometres. A 10 s swell wave travels at the speed \(c_0 \approx 1.56 T = 1.56 \times 10 = 15.6\ m/s = 56\ km/h\). The group velocity will be about half of that (in deep water). In Fig. 3.13 a graph is shown with wave height and period as a function of propagation distance from the storm centre.
At first order, no mass transport is associated with short wave propagation such that the path of swell through the oceans is unaffected by the Coriolis effect. Swell therefore travels the globe along great circles, the shortest distance between two locations on a spherical object.
Some coasts around the world – for instance Australia – mainly experience swell waves, which have been generated in storms far away. A typical wave spectrum will then be narrow-banded. For other coasts, locally generated storm waves dominate the wave climate, as is the case for the Dutch coast. The sea state then is irregular and short-crested. Most of the times wave records off the Dutch coast show both swell waves as generated in distant storms and storm waves locally generated. Two distinct peaks can then be observed in the spectrum, see Fig. 3.14. The swell can only come from the north and is usually not older than a day or sometimes two days.
3. In addition to dissipation and dispersion, non-linear wave transfer plays a role; energy is moved from the centre of the spectrum to both the higher and lower frequencies. The higher frequencies get sub- sequently dissipated, whereas the lower frequencies gain energy.