G58 



Miss M. Taylor on the Emission of Sound by a 



is emitted by a unit source in free space is Pep/Sir; so that 

 the presence of the cylindrical wall alters the total emission 

 ol energy in the ratio 2/k 2 a 2 . 



Next suppose that the period of the forced vibrations is 

 less than the pth free period but greater than the {p + l)th, 

 so that the first p values of m s 2 are negative, say m 2 = 

 The solution (15) may then be written 



Ps 2 . 



<f> 



2iirh 



v 



i 'Zira 



+ 2 



J (k s r)e- iu s* 

 HfM s {J Q (k s a)} 2 



Jo(k s r)e- m s% 1 



ikct . 



p+i 27ra 2 m s {J (k s a)Y 



■ (19) 



or, in real form, 



tirka 1 



_ sin k(ct—z) p J Q (fc s r) sin (kct - ul s z) 

 ' ^ + f 27ra 2 p s {J (k s a)} 2 



2 J (k 8 r)e- m ^ cos kct m 



p+ i 27ra 2 m s {J (k s a)} 2 



(20) 



so that the disturbance no longer reduces to a plane wave for 

 large values of z. 



Now suppose that the forced period is only very slightly 

 less than the pth free period, so that p p is very small. The 

 term into which this quantity enters will then be very great, 

 and will tend to infinity as p p tends to zero. The velocity 

 potential will of course also tend to infinity as the forced 

 period approaches coincidence with a free period from a 

 slightly greater value. There is, however, a remarkable 

 difference between the emission of energy in the two 

 cases. 



The energy given out by the source is obtained by inte- 

 grating ~pd<f>fdz over the section of the cylinder, where 

 p denotes the pressure of the fluid. Remembering that 

 p = Po + p~d < $>/'dt> where p denotes the undisturbed pressure 

 and p the density, the mean value of the energy emitted per 

 unit time is readily found from equation (20) to be 



P c 1 j? P kc 



. %ira 2 i S7rd 2 fji s { J [k]a) } 2 ' 



Thus, if the forced period is only slightly less than the /7th 

 free period, so that fi p . is very small, the expression for the 

 energy emitted becomes very great and tends to infinity as 

 fi P tends to zero. If, however, the forced period is slightly 



