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XLII. On the Momentum and Pressure of Gaseous Vibrations, 

 and on the Connexion with the Virial Theorem. By Lord 

 Eayleigh, O.M., F.R.S.* 



IN a paper on the Pressure of Vibrations (Phil. Mag. iii. 

 p. 338, 1902) I considered the case of a gas obeying 

 Boyle's law and vibrating within a cylinder in one dimension. 

 It appeared that in consequence of the vibrations a piston 

 closing the cylinder is subject to an additional pressure whose 

 amount is measured'by the volume-density of the total energy 

 of vibration. More recently, in an interesting paper (Phil. 

 Mag. ix. p. 393, 1905) Prof. Poynting has treated certain 

 aspects of the question, especially the momentum asso- 

 ciated with the propagation of progressive waves. Thus 

 prompted, I have returned to the consideration of the subject, 

 and have arrived at some more general results, which how- 

 ever do not in all respects fulfil the anticipations of Prof. 

 Poynting. I commence with a calculation similar to that 

 before given, but applicable to a gas in which the pressure 

 is any arbitrary function of the density. 



By the general hydrodynamical equation (Theory of 

 Sound, § 253 a), 



_Cdp_ dj> iTJ2 m 



-j — - - dt 2 U .... [1), 



where p denotes the pressure, p the density, $ the velocity- 

 potential, and U the resultant velocity at any point. If 

 we integrate over a long period of time, (/> disappears, and 

 we see that 



^dt + ^Wdt (2) 



retains a constant value at all points of the cylinder. The 

 value at the piston is accordingly the same as the mean value 

 taken over the length of the cylinder. 



If p u p Y denote the pressure and density at the piston, and 

 _p , p the pressure and density that would prevail throughout 

 were there no vibrations, we have 



P=f(p)=f(Po+P-po) .... (3), 

 and approximately 



= ^f /' (ft) + ^F p # {fH, f" M -f(Po) } ■ « • 



* Communicated bv the Author. 



OT 



