412 SURFACE WAVES. [CHAP. IX 



momentum in the direction of wave-propagation. The momentum, per wave 

 length, of the fluid contained between the free surface and a depth h (beneath 

 the level of the origin) which we will suppose to be great compared with X, is 



since ^ = 0, by hypothesis, at the surface, and ch, by (1), at the great depth 

 h. In the absence of waves, the equation to the upper surface would be 

 y=^a 2 , by (3), and the corresponding value of the momentum would there 



fore be 



A .................................. (v). 



The difference of these results is equal to 



7rpa 2 c ....................................... (vi), 



which gives therefore the momentum, per wave-length, of a system of 

 progressive waves of permanent type, moving over water which is at rest at a 

 great depth. 



To find the vertical distribution of this momentum, we remark that the 

 equation of a stream-line ty = cti is found from (2) by writing y-f- h for ?/, and 

 j3e~ kh for /3. The mean-level of this stream-line is therefore given by 



y= -h + ^^e~ 2kh ........................... (vii). 



Hence the momentum, in the case of undisturbed flow, of the stratum of 

 fluid included between the surface and the stream-line in question would 

 be, per wave-length, 



pcAtA +JJ/S* (!--&quot;)} ....................... (viii). 



The actual momentum being pcA X, we have, for the momentum of the same 

 stratum in the case of waves advancing over still water, 



7rp 2 c(l-e- 2fc &amp;gt;0 ................................. (ix). 



It appears therefore that the motion of the individual particles, in these 

 progressive waves of permanent type, is not purely oscillatory, and that there 

 is, on the whole, a slow but continued advance in the direction of wave- 

 propagation*. The rate of this flow at a depth h is found approximately 

 by differentiating (ix) with respect to A , and dividing by pX, viz. it is 



tfatce-* .................................... (x). 



This diminishes rapidly from the surface downwards. 



232. A system of exact equations, expressing a possible form 

 of wave-motion when the depth of the fluid is infinite, was given 

 so long ago as 1802 by Gerstnerf, and at a later period indepen 

 dently by Rankine. The circumstance, however, that the motion 



1 * Stokes, 1. c. ante, p. 409. Another very simple proof of this statement has 

 been given by Lord Eayleigh, 1. c. ante, p. 279. 



t Professor of Mathematics at Prague, 17891823. 



