217] STANDING WAVES. 373 



In terms of a we have 



ga cosh k (y + h) , / , \ /0 \ 



6 = Z -^ - cos lex . cos (at + e) (8), 



a- cosh kh 



and it is easily seen from Art. 62 that the corresponding value of 

 the stream-function is 



ga sinh k (y + h) . 7 ( N /nx 



-\lr=^ r^-5 sin &a? . cos (at 4- e) (9). 



o- cosh M 



If x, y be the coordinates of a particle relative to its mean 

 position (x, y), we have 



~dt = ~d~x ) ~dt = ~Ty (10) ' 



if we neglect the differences between the component velocities at 

 the points (x } y) and (x + x, y + y), as being small quantities of 

 the second order. 



Substituting from (8), and integrating with respect to t, we find 



x = a . .; y 7 sni kx . sin (at + e), 



sinn Kit 



Y = a . ,,, ' cos ^ . sm (at + e) 



smh M 



where a slight reduction has been effected by means of (5). The 

 motion of each particle is rectilinear, a ad simple-harmonic, the 

 direction of motion varying from vertical, beneath the crests and 

 hollows (kx=m7r), to horizontal, beneath the nodes (kx = (m + ) TT). 

 As we pass downwards from the surface to the bottom the ampli- 

 tude of the vertical motion diminishes from a cos kx to 0, whilst 

 that of the horizontal motion diminishes in the ratio coshM : 1. 



When the wave-length is very small compared with the 

 depth, kh is large, and therefore tanh kh = l. The formulae (11) 

 then reduce to 



x = a^ y sin kx . sin (crt + e), \ ... ~, 



y= ad^ cos kx . sin (at + e) j 



with <r* = gk (13)* 



The motion now diminishes rapidly from the surface down- 

 * This case may of course be more easily investigated independently. 



