Lord Rayleigh on Acoustic Repulsion. 271 



the velocity is negligible, but, on the other hand, the variation 

 of pressure considerable. It follows that 



j"Oi-O* = 0, (4) 



or that the mean value of w in the interior is the same as at a 

 distance outside. 



The remainder of the investigation depends upon the rela- 

 tion between p and p. If the expansions and contractions are 

 isothermal, p = a 2 p, and iv=a 2 log p. Thus 



§logp 1 dt=logp Q .t; (5) 



or the mean logarithmic pressure in the interior is the same as 

 the constant logarithmic pressure at a distance. Equation (5) 

 may also be written 



J]^(i+£=a)*=o, .... ( 6) 



or 



_J , { e ?-'(°?) ,+ ">-'"- • « 



whence, if the changes of pressure be relatively small, we see 

 that the mean value of pi— p is positive, or, in other words, 

 that the mean pressure inside the resonator is in excess of the 

 atmospheric pressure. 



If, as in practice, the expansions and contractions are adia- 

 batic, p &-p y , where 7=1*4, and (5) is replaced by 



y-i Y-i 

 py dt=p Q y..t (8) 



Thus, instead of (7), 



whence, by the binomial theorem, 



£?t Aft ?)* •■••■'• <*» 



approximately, showing that here again the mean pressure in 

 the interior of the cavity exceeds the atmospheric pressure. 

 Hence, on either supposition, the resonator tends to move as 

 if impelled by a force acting normally over the area of its aper- 

 ture and directed inwards. 



I am, Gentlemen, 



Your obedient Servant, 



Rayleigh. 

 Terling Place, Witham, 

 September 7, 1878. 





