44:0 Calculations on Momentum of Progressive Waves. 



as the generalized form of (20). It is equivalent to a 

 relation given first in another notation by De Morgan *, 

 and it may be regarded as the condition which must be 

 satisfied if the emergence of a negative wave is to be 

 obviated. 



- We are now prepared to calculate the momentum. For 

 a wave in which the mean elevation is zero, the momentum 

 corresponding to unit horizontal breadth is 



p§u(h + v)dx = W(9l h o)$y^> • • (22) 



when we omit cubes and higher powers of rj. We may 

 write (22) also in the form 



Momentum = | T ° tal f lerS ' V , . . . (23) 



a being the velocity of propagation of waves of small 

 elevation. 



As in (14) y with 7 equal to 2, we may prove that the 

 momentum is positive without restriction upon the value 



Of 7]. 



As another example, periodic waves moving on the surface 

 of deep water may also be referred to. The momentum of 

 such waves has been calculated by Lamb f , on the basis of 

 Stokes' second approximation. It appears that the momen- 

 tum per wave-length and per unit width perpendicular to the 

 plane of motion is 



irpa\ (24) 



where c is the velocity of propagation of the waves in 

 question and the wave form is approximately 



97 = a cos— -(ct — 3d) 



(25) 



The forward velocity of the surface layers was remarked 

 by Stokes. For a simple view of the matter reference may 

 be made also to Phil. Mag. vol. i. p. 257 (1876) ; Scientific 

 Papers, vol. i. p. 263. 



* Airy, Phil. Mag. vol. xxxiv. p. 402 (1849). 

 f Hydrodynamics, § 246. 



