﻿9P' 



Dr. J. E. Wilton on Ripples. 689 



in which p is the density of water, p' of the air, T is the 

 surface tension, and the rest of the notation is that of 

 " Waves." 



It is assumed in obtaining the above value of c 2 that the 

 air is incompressible, but the removal of this restriction 

 will rather lessen than increase the effect of air waves. 



Since p'/p is small the effect of p' is to multiply one term 

 in c 2 by 12 — p'/p, the other by I— p'/p. Now 



^ = •0013 



nearly. Hence the correction is of the order 1/400 of the 

 uncorrected value. We shall consider this as negligible. 

 In fact, in such a (relatively) high wave as that of " Waves/' 

 fig. 1, the ratio of the last term retained to the first is of 

 the order 



A 12 /A 1= = 1/150. 



If, then, we omit p', we have as a first approximation 



2?rc 2 _ /2tt\ 2 T 

 gX + \X) 



which we shall, for brevity, write in the form 



fJL=l + /C. 



But 



T/gp = 74/981 = -075, 



so that tc, the correction due to surface tension, is appreciable 

 for waves of length less than about 25 cm., and for very 

 short ripples it may become very large. 



Finally, we have to take account of viscosity. It is shown 

 in Lamb's 'Hydrodynamics,' § 332, p. 566 (Third Edition), 

 that, if X/'0048 cm. may be considered large (say 10 or 

 more), the effect of viscosity is to introduce a time factor 

 e -2v(2Tr/A)2t w hich does not affect the form of the wave, ami to 

 introduce a correcting factor to the form of the wave of the 

 order of magnitude 



2^( 2tt /A.) 2 _ 2^/-^(2tt/X)1 

 /gX , 2?rT " x/T+K ' . 

 V 2tt + X p 



v being the coefficient of dynamical viscosity. 



Phil. Mag. S. 6. Vol. 29. No. 173. May 1915, 2 V 



