the Breaking of Waves. 359 



and for the reflected wave 



., cosh My — h) 7/ , A 



6 =c . , , , — -cosk(x + ct). 



T Sinn kk 



Hence we take the velocity system : 



, cosh&(// -h) . , 



«' = — kc • . /7 — - sm A; (a? -f ct), 



sinh A;/i 



, , sinh k(y — h) 7/ N 



'^" imtit cos *(* + <*), 



leading at the origin to the equation : 



- 9R = n— m) coth kh . ck 2 cos & ct T 

 p ot 



whence 



(1— >»)£ coth M sin tcf 



P ro' 



(ni^l) . coth kh . 2irnjK 



— Po e 5 



so that the tendency to rupture now depends on rfcoihk7i/\ 

 or 7]jc 2 . Since coth kh > 1 we see that, for a given wave- 

 length, waves break more readily in shallow water than in 

 deep. 



If we examine the formulae, we find that they agree fairly 

 well with the behaviour of waves breaking against a sea- 

 wall or pier. On a quiet day, the waves meeting the wall are 

 reflected in the ordinary way, but when the waves are large 

 enough to exceed the critical value of r], they break. An 

 advancing wave causes the water to rise smoothly against the 

 wall up to a certain point, when, for no apparent reason, 

 it breaks suddenly into spray. This is what the formulae 

 lead us to expect, for when once the critical value of rj is 

 reached, p/p increases as an exponential function of rj, so 

 that the transition from smooth motion to spray is rapid. 

 When the trough of the wave reaches the wall, the water 

 sinks quietly to a certain point, and then begins suddenly to 

 seethe, the critical value of t] being again passed. 



Since y is measured downwards it is negative at the crests 

 of the waves. Now m — 1 is negative, hence p>p at the 

 crests, and the spraying is due to the rapid rise of pressure 

 consequent on the inability of the water to contract in 

 accordance with the formula. The seething in the troughs 

 is similarly due to water running down the slopes of the 

 waves to supply the deficiency due to the inability of the 

 water to expand, as required by the fact that p < p . 



The slope of the exponential curve being greater for positive 



2 B 2 



