Figure 5. Illustrating the two kinds of breaking of water waves. Curve (a): wave of greatest height 

 in deep water; further increase of wave energy produces spray at the crests. 

 Curve (b): amplitude dispersion converts this sinusoidal wave on shallow water in a very short 

 distance of travel into the breaking form (c) and then into a succession of bores. 



full advantage. The observer must be stationed nearly in a line with the ridges of 

 the waves where they begin to break." Stokes calculated also the form of waves of 

 intermediate height, which are usually called Stokes waves (see Fig. 4). 



10. Amplitude dispersion 



By contrast, at the long-wave end of the scale, say for A >8/z, the existence 

 of periodic waves is limited by amplitude dispersion. By this I mean the tendency for 

 each value 77 of surface elevation to be propagated with the speed ~\/grj appropriate 

 to the local depth rather than with the speed \/gh corresponding to the mean depth. It 

 is well known how this* leads to steepening of the front of the wave until it may 

 become a "hydraulic jump" or "bore." 



11. Cnoidal waves and the solitary wave 



However, if the amplitude is not too great, this amplitude dispersion may be 

 exactly balanced by frequency dispersion in such a way that periodic waves become 

 possible. These are the "cnoidal" waves of Korteweg and de Vries [17], so called 

 because their waveform is the square of the Jacobian elliptic function known as "en." 

 Their existence depends on the fact that the higher harmonics in the wave travel 

 slightly slower than the fundamental, which for particular waveforms can just balance 

 the steepening just mentioned. Changes due to this frequency dispersion are propor- 

 tional to h 2 /\ 2 , but changes due to amplitude dispersion are proportional to a/h; as a 

 result, any given cnoidal shape is possible for a fixed ratio (a/h) : (h 2 /\ 2 ). The 

 limiting case here is the solitary wave with a\-/h 3 ~ 25. For amplitudes above that 

 of the solitary wave we must have steepening and bore formation. For amplitudes 

 just below it the cnoidal wave train is beginning to degenerate into a sequence of 

 isolated solitary waves. 



* Together with the fact that the particle velocity is forwards where 77 is greatest and 

 backwards where i\ is least. 



24 



