﻿694 Dr. J. B. Wilton on Ripples. 



It will be found that equations (10), . . . (15) agree with 

 (16) and (17), and that they also agree with the known 

 result when /e = 0. Hence it is improbable that there is any 

 undetected error in calculation. 



The most interesting thing about the constants whose 

 values are given in equations (10), . . . (15) is the unexpected 

 form of the denominators. It is easy to see that this form 

 is general ; for the coefficient of A n in the equation which 

 determines it is 



2nQ + 1 + n 2 K = n 2 /c + 1 - w(l + k) 



= (n — l)(n/c— 1), 



to a first approximation. Hence the denominator of A^ 



n 



contains as a factor (n — 1) ! n (tk—1). Now we have seen 



that we are justified in neglecting the effect of air waves 

 if k is greater than about *01 ; so that for a considerable 

 range of values of n we have to consider the possibility 

 of values of k of the form K=l/n, where n is a positive 

 integer. To the consideration of these values of k we shall 

 return later. On the other hand, since we are not justified 

 in neglecting the effect of air-waves if k is less than about 

 •01, we cannot, as might appear at first sight from equations 

 (10), . . . (15), conclude that however small k is, so long as it 

 is finite, the value of one of the coefficients A n becomes 

 illusory, and therefore that the ordinnry theory, in which 

 k = 0, is incorrect. 



Let us take first the case of a ripple whose form is largely 

 determined by surface tension — say that for which k=10, 

 and therefore A,= *54 cm. From equations (10), . . . (15) 

 I find, for this ripple, 



Ax=— a, A 2 = '21a 2 - '007 2a\ 



A 3 = - -033a 3 + -0049a 5 , A 4 =*0031a 4 , A 5 = -00023a 5 , 



/*=ll-l-43a 2 + '014a 4 . 



The largest value of a which we may safely insert in these 

 equations is * a = 1'5. We then find 



A 1= -l-5, A 2 = -44, A 3 =--075, 



A, = *016, A 5 = -0017, 



^, = 7*85, i. e., c=25'7 cm./sec, 



X = -54 cm. 



* It may be verified that R is positive for this value of a when £=0. 



