183-184] WAVES OF FINITE AMPLITUDE. 299 



differs from zero therefore advances, whilst that within which Q 

 differs from zero recedes, so that after a time these regions 

 separate, and leave between them a space within which P = 0, 

 Q = 0, and the fluid is therefore at rest. The original disturbance 

 has now been resolved into two progressive waves travelling in 

 opposite directions. 



In the advancing wave we have 



(12), 



so that the elevation and the particle- velocity are connected by a 

 definite relation (cf. Art. 168). The wave- velocity is given by (10) 

 and (12), viz. it is 



To the first order of 7j/h, this is in agreement with Airy s result. 



Similar conclusions can be drawn in regard to the receding 

 wave*. 



Since the wave-velocity increases with the elevation, it appears 

 that in a progressive wave- system the slopes will become con 

 tinually steeper in front, and more gradual behind, until at length 

 a state of things is reached in which we are no longer justified in 

 neglecting the vertical acceleration. As to what happens after 

 this point we have at present no guide from theory ; observation 

 shews, however, that the crests tend ultimately to curl over and 

 break. 



184. In the application of the equations (1) and (3) to tidal 

 phenomena, it is most convenient to follow the method of successive 

 approximation. As an example, we will take the case of a canal 

 communicating at one end (% = 0) with an open sea, where 

 the elevation is given by 



rj = a cos crt 

 For a first approximation we have 



du dn dn 1 du 



- 



* The above results can also be deduced from the equation (3) of Art. 170, to 

 which Biemann s method can be readily adapted. 



