432 



SWAN. 



x - ae^ kvt+v ) 



dcp dx 



— = — = ikvae« kvt+v) 



dn dt 



<Px 

 dt" 



= - *Waa«** f+ *> 



Then 



dW ira dx d 2 x 



dx 

 dt 



1 - 



Ji(2feo) ' 



ka 



dx drx 



Taking the real part of — and -rr , we have 



dt dt~ 



7Tl 3 (Xa 



dW 



-j- = sin {kvt + 7?) cos {kvt + v) Ki(2ka) 



-f- irarkh^cra" sin (kvt + 77) 



1 - 



Jii'Ika) 

 ka 



The first term is entirely periodic and hence disappears after the 

 fork has made a few vibrations and gained its full amplitude. The 

 mean value of sin 2 {kvt + 77) is J. Hence, in the steady condition, 



W = 



ira"k"v 3 crcct 



1 - 



JiMa) 

 ka 



Hence the average rate of emission of energy is 



E = — = 2ir 3 arr 



O 9 



07m- 



vJ\ 



1 - 



47T/?fif 



lirna 



Sabine 21 has shown that in a closed room of volume V, 



E= VAi' e Ai 

 where i' is the minimum audible energy per cubic unit, and 



A = 



I 2 



loge 7- loge 



a* 



Oil 



h - U 



h - h 



21 loc. cit. 



