i 84 Lord Rayleigh : Hydrodynamical Notes. 



Stokes *. In a supplement published in 1880 f the same 

 author treated the problem by another method in which the 

 space coordinates x, y are regarded as functions of </>, -v^ the 

 velocity and stream functions, and carried the approximation 

 a stage further. 



In an early publication % I showed that some of the results 

 of Stokes's first memoir could be very simply derived from 

 the expression for the stream-function in terms of x and y, 

 and lately I have found that this method may be extended 

 to give, as readily if perhaps less elegantly, all the results of 

 Stokes' supplement. 



Supposing for brevity that the wave-length is 2tt and the 

 velocity of propagation unity, we take as the expression for 

 the stream-function of the waves, reduced to rest, 



ty=y — a.e~ y cosx—jBe~ 2y cos 2x - ye~ Sy cos 3#, . (1) 



In which x is measured horizontally and y vertically 

 downwards. This expression evidently satisfies the differ- 

 ential equation to which yjr is subject, whatever may be the 

 values of the constants a, p, 7. From (1) we find 



U 2 -2yy = (dylr/dxy+(df/dyy-2 9 y 



= l-2ylr + 2(l^g)y + 2/3e- 2 Vcos2x + Aye- 2 ycos3x 



+ <x?e- 2 y + 4/3 2 e-*y + 9y 2 e- 6 y+ 4:u0 e-s* 60s x 



+ 6aye-^cos2x + 120ye-^cos,v (2) 



The condition to be satisfied at a free surface is the constancy 

 of (2). 



The solution to a moderate degree of approximation (as 

 already referred to) may be obtained with omission of /3 and 

 7 in (1), (2). Thus from (1) we get, determining yjr so that 

 the mean value of y is zero, 



t/ = a(l + |a 2 ) COS.Z — J a 2 COS 2a' + fa 3 COS 3^, . . (3) 



which is correct as far as a 3 inclusive. 



If we call the coefficient of cos x in (3) a, we may write 

 with the same approximation 



y = acosx — ^a 2 cos2x + %a s cos3x. . . . (4) 



* Camb. Phil. Soc. Trans, viii. p. 441 (1847) ; ' Math, and Phys. Papers,' 

 i. p. 197. 



t L. c. i. p. 314. 

 . % Phil. Mag. i, p. 257 (1876) ; Sci. Tapers, i. p. 262. See also Lamb's 

 ' Hydrodynamics;' § 230. 



