Calculations on Momentum of Progressive Waves. 437 



I£ /3=1, the law of pressure is that under which waves 

 can be propagated without a change of type, and we see that 

 the momentum is zero. In general, the momentum is 

 positive or negative according as /9 is less or greater than 1. 



In the above formula (2) the calculation is approximate 

 only, powers of the disturbance above the second being- 

 neglected. In the present note it is proposed to determine 

 the sign of the momentum under the laws (3) and (6) more 

 generally and further to extend the calculations to waves in 

 a liquid moving in two dimensions under gravity. 



It should be clearly understood that the discussion relates 

 to progressive waves. If this restriction be dispensed with, 

 it would always be possible to have a disturbance (limited if 

 we please to a finite length) without momentum, as could be 

 effected very simply by beginning with displacements un- 

 accompanied by velocities. And the disturbance, considered 

 as a whole, can never acquire (or lose) momentum. In order 

 that a wave may be progressive in one direction only, a 

 relation must subsist between the velocity and density at 

 every point. In the case of Boyle's law this relation, first 

 given by De Morgan *, is 



u = a log (p/p ), (9) 



and more generally f 



-wm 



.... (10) 



Wherever this relation is violated, a wave emerges travelling 

 in the negative direction. 



For the adiabatic law (3), (10) gives 



-&{(£)*-<}•• ■ • • a.) 



a being the velocity of infinitely small disturbances, and this 

 reduces to (9) when 7 = 1. Whether 7 be greater or less 

 than 1, u is positive when p exceeds p . Similarly if the 

 law of pressure be that expressed in (6), 



•<^{'-(^'}- '■ ■ ■ m 



Since ft is positive, values of p greater than p are here 

 also accompanied by positive values of u. 



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

 f Earnshaw, Phil. Trans. 1859, p. 146. 



