249] 



WAVES AND RIPPLES. 



453 



It is worth notice that, in contrast with 

 the case of Art. 227, the elevation is now 

 finite when x = 0, viz. we have 





This follows easily from (10). 



The figure shews the transition between 

 the two sets of waves, in the case of K 2 = 5^. 



The general explanation of the effects 

 of an isolated pressure-disturbance advanc- 

 ing over still water, indicated near the end 

 of Art. 227, is now modified by the fact 

 that there are two wave-lengths correspond- 

 ing to the given velocity c. For one of 

 these (the shorter) the group-velocity is 

 greater, whilst for the other it is less, than 

 c. We can thus understand why the waves 

 of shorter wave-length should be found 

 ahead, and those of longer wave-length in 

 the rear, of the disturbing pressure. 



It will be noticed that the formulae (10), 

 (11) make the height of the up-stream 

 capillary waves the same as that of the 

 down-stream gravity waves ; but this result 

 will be greatly modified when the pressure 

 is diffused over a band of finite breadth, 

 instead of being concentrated on a mathe- 

 matical line. If, for example, the breadth 

 of the band do not exceed one-fourth of the 

 wave-length on the down-stream side, whilst 

 it considerably exceeds the wave-length of 

 the up-stream ripples, as may happen with 

 a very moderate velocity, the different parts 

 of the breadth will on the whole reinforce 

 one another as regards their action on the 

 down-stream side, whilst on the up-stream 

 side we shall have ' interference,' with a 

 comparatively small residual amplitude. 



