﻿Dr. J. R. Wilton on Ripples. 

 Table III. 

 Ripple o£ wave-length 1*31 cm. 



697 



s. 



c£— #. 



y- 











•077 



7T/6 



•068 



•068 



,r/3 



•144 



•031 



7T/2 



•258 



-•009 



2tt/3 



•389 



-•039 



5tt/6 



•522 



-054 



7T 



•653 



-•060 



Fig. 3. 



Ripple of length 1*31 cm. ; amplitude "14 cm. ; velocity 194 cm. /sec. 



We come now to the most interesting portion of our 

 inquiry, — the consideration of the form of those waves 

 for which k is the reciprocal of a positive integer n other 

 than unity. When k is not actually equal to ljn, it is 

 always possible to choose a sufficiently small to insure 

 the convergence of the series for A 2 , A 3 , .... For it is 

 easy to satisfy oneself that the index of the power of uk — 1 

 in the denominator of any coefficient is less than the index 

 of the power of a which it multiplies. Hence if we put 

 a=(n/c— 1)&, some power of nic—1 will divide every co- 

 efficient A m , and it is now manifestly possible to choose 

 a value of b which secures convergence of the series. If 

 nfc — l is small a will be small, i. e. the amplitude of the 

 wave will be small ; but as A„ and the succeeding co- 

 efficients become relatively important the form of the wave 

 may be very different from that of a simple cosine curve. 



When K=ljn the case is different. The ordinary method 

 of approximation breaks down altogether, and we have to 

 start again ab initio. We shall consider in particular the 

 case /c = h 



When tc — -J, to a first approximation C= — f, and therefore 

 equation (6) leads to 



|A] 2 + terms of order higher than the second = 0. 

 Hence Ax cannot be of the first order unless A L > is of the 



