662 Miss M. Taylor on the Emission of Sound hy a 



Now suppose that the period of the forced vibration co- 

 incides with one of the periods of free radial vibration in the 

 case of no friction, say k = k p . Then ot P = ^p = \/(kfij2c), 

 and the corresponding term in the velocity potential, ex- 

 pressed as a real quantity, is 



e - Wp/te)*j Q lkr){coa (kc t-y/(k/il2c)z) + sin (kct — ^/(k/i/2c)z) 



4*raV(*M/2c){Jo(*/0}' 



.... (40) 



This is finite, even in the case of isochronism, though it 

 tends to infinity as pu tends to zero. 



To find the energy emitted by the source we must take 

 the integral of the expression —pb<j>fdz over the cross- 

 section z = 0. We find, from (36) 



_(M\ _± 



? * , ^ J o(k 8 r) cos kct 

 cos kct + 2 o » tt M2 



27ra-{J [k a a } 2 



Also 

 Hence 



V =Po + c 2 ps =p + /o(iAc + /x) 0. 

 fa cos &itf + /3 sin &6'/j 



(41) 



+2 



V+/3 2 ) 



J (h,r)(*B cos te + /3 S sin ?.-<•/) 



] 



+ 



W(«,* + &'HJ,(*.«)} S 



, r/3 cos £rf — a sin farf _, J (k s r)(fi s c6skct~ ««sin£ci)~l 



.... (42) 

 Hence the mean value of the energy emitted per unit time is 



PflCL 



+ 



pkcl 



S7nr(a 2 +./3 2 ) S7ra 2 (a 2 + /3 2 ) 



oT + 2, 



p(fj,x s + kc/3 s 



87ra 2 (a s 2 + ft 2 ){Jo(/^)} 2 ' 

 .... (43) 



If the forced period differs only slightly from the pth free 

 period, so that k 2 — k 2 is small, we can write 



, . . (44) 



-/3 2 = /L 2 -P==h- 



±6 



V •> 



the upper or lower sign being taken according as the forced 

 or free period is the greater. Also 



2^/%, =*/*/<■. 



