concerning the Momentum of Progressive Waves. 439* 



and negative if /3is greater than 1. If/3 = 1, the momentum, 

 vanishes. The conclusions formerly obtained on the sup- 

 position of small disturbances are thus extended. 



We will now discuss the momentum in certain cases of 

 fluid motion under gravity. The simplest is that of long 

 waves in a uniform canal. If rj be the (small) elevation at 

 any point x measured in the direction of the length of the 

 canal and u the corresponding fluid velocity parallel to ./•,. 

 which is uniform over the section, the dynamical equation is * 



du dn 



jt=-e£ ( 17 > 



As is well known, long waves of small elevation are pro- 

 pagated without change of form. If c be the velocity of 

 propagation, a positive wave may be represented by 



v =F(ct-.v), (18) 



where F denotes an arbitrary function, and c is related to 

 the depth h according to 



c 2 =gh (19) 



From (17), (18) 



-?V©." ™ 



is the relation obtaining between the velocity and elevation 

 at any place in a positive progressive wave of small 

 elevation. 



Equation (20), however, does not suffice for our present 

 purpose. We may extend it by the consideration that in a 

 long wave of finite disturbance the elevation and velocity 

 may be taken as relative to the neighbouring parts of the 

 wave. Thus, writing du for u and h for 7i 0) so that rj = dJu 

 we have 



du= \/($) dh >- 



and on integration 



The arbitrary constant of integration is determined by the 

 fact that outside the wave k = when h = h , whence and 

 replacing h by Iiq + t], we get 



««V*{v'(*o+»)-v'M. • • • ( 21 > 



* Lamb's Hydrodynamics, § 168. 



