[ 231 ] 



XX. Note on the Theory of the Double Resonator. 

 By lord Rayleigh, O.M., F.R.S.* 



N my book on the ' Theory of Sound ' f I have con- 

 sidered the case of a double resonator (fig. 1), where 



Fiff. 1. 



two reservoirs of volumes S, S' communicate with each 

 other and with the external atmosphere by narrow passages 

 or necks. If we were to treat SS' as a single reservoir 

 and apply the usual formula, we should be led to an 

 erroneous result ; for that formula is founded on the 

 assumption that within the reservoir the inertia of the air 

 may be left out of account, whereas it is evident that the 

 energy of the motion through the connecting passage may 

 be as great as through the two others. However, an investi- 

 gation on the same general plan meets the case perfectly. 

 Denoting by X 1? X 2 , X 3 the total transfers of fluid through 

 the three passages, we have for the kinetic energy the 

 expression 



" 2p \cA dt / + Adt ) "*" c z \dt J / ' 



and for the potential energy 



v-w{£> 



Xrf , (X 3 -X 2 ) ! 



s 



0) 



• (2) 



Here p denotes the density of the fluid, a the velocity of 

 sound, while c l5 c 2 , c 3 may be interpreted as the electrical 

 conductivities of the passages. Thus for a long cylindrical 

 neck of radius R and length L we should have c = 7rRyL. 



* Communicated by the Author. 



f § 310, first edition 1878, second edition 18£6, Macmillar. Also 

 Phil. Trans. 1870; Scientific Papers, vol. i. p. 41. 



