PIPES AND RESONATORS 275 



source in unison with it may be sufficiently illustrated on the 

 assumption that the incident waves are plane. If 



fa = Jcosk(ct-x), ............... (22) 



the ratio of the energy scattered by the resonator, which is 

 given by (12), to the energy-flux in the primary waves, viz. 

 JpAr'cJ" 2 , is 47T/& 2 , or X 2 /7r. The energy diverted per second, 

 at its maximum, is therefore equal to '318 of that which in 

 the primary waves is transmitted across a square area whose 

 side is the wave-length. It may be added that a similar 

 law is met with in the theory of selective absorption of 

 light. 



When approximate agreement between the frequency 

 (n/2?r) of the source and the natural frequency (w /27r) of the 

 resonator is no longer assumed, the external pressure which is 

 required to maintain a steady vibration (1) through the aperture 

 will consist of two parts. In the first place we have a component 

 keeping step with the displacement, which is required in order 

 to control the inertia of the air. This is easily found by an 

 extension of the method of 86. If $1 denote the velocity- 

 potential in the interior of the resonator, </> 2 that at a short 

 distance outside the aperture, in the region of approximately 

 spherical waves, we have 



q-Kfa-M .................. (23) 



in accordance with the electrical analogy. In the interior we 

 have 5 = q/Q, c 2 s = ^, as before. Hence 



q + n<?q = -K<f> 2 , .................. (24) 



where n 2 = ^Tc 2 /Q ...................... (25) 



This gives, for the external pressure, 



(26) 



The second part, which is in the same phase as q, is needed in 

 order that there may on the average be no gain or loss of energy 

 to the air contained in the resonator, and is accordingly given 

 by (7). Hence we have, altogether, 



-smnt); ...(27) 

 182 



