PIPES AND RESONATORS 265 



If the resonator were practically isolated in space, then on 

 account of the assumed smallness of its dimensions as com- 

 pared with \, the effect of the flux at a distance would be that 

 of a simple source of strength nC, and the rate of emission of 

 energy would accordingly be 



W=n 4 pC*/Sirc, ..................... (14) 



by the formula (15) of 76. The energy E of the motion, 

 being equal to the potential energy at its maximum, is, 

 approximately, 



(15) 



by (12). Equating, on the principles of 11, the rate of decay 

 of this energy to the emission W, we find 



and therefore q = G e~ tlr cos (nt + e), ............... (17) 



provided r = $7r(?/n'Q = S-rrQ/K^ ............ (18) 



in virtue of (9). The ratio of the modulus of decay to the 

 period (%7r/kc) is given by 



(19) 



Since K is at most comparable with the mean breadth of the 

 aperture, this ratio is usually very great, and the preliminary 

 assumptions implied in the above process are amply justified. 



If the mouth of the resonator were furnished with an 

 infinite flange, i.e. one whose breadth is large compared with X, 

 the equivalent source would, as explained in 82, have double 

 the strength above assumed, and the emission of energy, now 

 operative in one half of the surrounding region, would be twice 

 as great. The modulus (18) would accordingly be halved. 



As a numerical illustration of the theoretical results, take 

 the case of a spherical vessel 10 cm. in diameter, with a circular 

 aperture 1 cm. in radius, so that Q = 523*6, K 2. The wave- 

 length, calculated from (10), is 101*6 ; and the frequency there- 

 fore about 327. The modulus of decay, as given by (18), is 

 about one-tenth of a second. 



