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XLVII. Further Calculations concerning the Momentum of 

 Progressive Waves. By Lord Rayleigh, O.M., F.R.S.* 



THE question of the momentum of waves in fluid is of 

 interest and has given rise to some difference of opinion. 

 In a paper published several years ago f I gave an approxi- 

 mate treatment of some problems of this kind. For a fluid 

 moving in one dimension for which the relation between 

 pressure and density is expressed by 



p=Kp) (i) 



it appeared that the momentum of a progressive wave of 

 mean density equal to that of the undisturbed fluid is given 



{^^ + i} xt0talener ^- • • ( 2 > 



m which p is the undisturbed density and a the velocity of 

 propagation. The momentum is reckoned positive when it 

 is in the direction of wave-propagation. 

 For the "adiabatic" law, viz. : 



P/Po=(p/poy, ........ (3) 



/ W = fW, / >0 ) = ^fl), . . (4) 



., , Po Po 



so that 



PofiPo) 1 _y+l ... 



4a 3 T 2a " 4a W 



In the case of Boyle's law we have merely to make 7=1 

 in (5). 



For ordinary gases y>l and the momentum is positive; 

 but the above argument applies to all positive values of 7. 

 If 7 be negative, the pressure would increase as the density 

 decreases, and the fluid would be essentially unstable. 



However, a slightly modified form of (3) allows the 

 exponent to be negative. If we take 



p/p =2-(p/ Po )-i> (6) 



with ft positive, we get as above 



/fo)=&W, /"W^M+^i . (7 ) 



and accordingly 



pj"(p ) 1 _ 1-/3 



4a 3 i "2a" 4a W 



* Communicated by the Author. 



t Phil. Mag. vol. x. p. 364 (1905) ; Scientific Papers, vol. v. p. 26-5. 



