217] STANDING WAVES. 373 



In terms of a we have 



ga cosh k (y + h) , , , , /CA 



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



cr cosh kh 



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

 the stream-function is 



qa sinh k (y -f h) . 7 , . &amp;gt;. /m 



Jr = v -- 2 &amp;gt; sm kx . cos (at + e) ......... (9). 



or cosh kh 



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

 position (x, y), we have 



(ft~ cfo d* dy&quot; 



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 



cosh k (y + h) . . . , x 



x. = a - . , . . - sm KX . sm (at 4- e), 

 sinh kh 



y = a -- . U ; , - cos kx . sin (cr^ + e) 

 sinh kh 



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

 motion of each particle is rectilinear, and simple-harmonic, the 

 direction of motion varying from vertical, beneath the crests and 

 hollows (kxm jr)^ to horizontal, beneath the nodes (kx = (m + J) 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 cosh kh : 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 = - ae*y sin &# . sin (&amp;lt;r + e), 



( . 



y = ae 1 cos kx . sin (crt + e) } 



with a^ = gk ........................... (13)* 



The motion now diminishes rapidly from the surface down- 



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



