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 



(iv), 



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

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

 y=4a 2 , by (3), and the corresponding value of the momentum would there- 

 fore be 



pc(A+pa')X .................................. (v). 



The difference of these results is equal to 



irpa 2 ? ...................................... .(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 -^ ch' is found from (2) by writing y-\- h' fory, and 

 p e -kh' f or ^ TI^ mean-level of this stream-line is therefore given by 



y= -A' + i^e-a* ........................... (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, 



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

 stratum in the case of waves advancing over still water, 



7rpa*c(l-e- zkh ') ................................. (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 A' is found approximately 

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



Watce-M .................................... (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 



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

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



t Professor of Mathematics at Prague, 17891823. 



