PIPES AND RESONATORS 273 



under the condition of maximum resonance, when n is given 

 by (3) approximately. The corresponding flux is 



g = -g-sin(?rt-e) ................... (11) 



The emission of energy is best calculated from a con- 

 sideration of the circumstances at a great distance. The 

 velocity-potential will be compounded of that due to the 

 original source and that due to the flux q, and under certain 

 conditions the latter component may greatly preponderate. 

 The emission of energy is then 



W=27rpcJ*, ..................... (12) 



approximately, by 76 (15). 



Thus if < 2 be due to a simple source A cos kct at a distance b 

 from the aperture, we have 



&) ................ (13) 



Hence J=A/4nrb, 



and q = j-j sin k (ct b) ................... (14) 



This is equivalent to a source whose amplitude is to that of 

 the primary source in the ratio l/kb. If b be small compared 

 with X/27T this ratio is large ; and the emission of energy 

 exceeds that due to the original source in the ratio l/k?b z . 

 In the case of a double source B cos kct we may write, 

 if kb be small, 



D 



< 2 = 7 Tg c s a cos k(ctb), ............ (15) 



by 76 (23), if a denote the angle which the axis of the source 

 makes with the line drawn from it to the aperture. Hence 

 / = B cos a/4?r6 2 , and the emission, as given by (12), is 



TF = pc J B 2 cos 2 a/87r& 4 ................ (16) 



The emission due to the original (double) source alone would 

 be ptfcIP/ZiTr, by 76 (26). The ratio in which the emission 

 is increased is therefore 3 cos* a/fab*. Since the mean value 

 of cos 2 a is J, the mean value of this ratio, for all directions 

 of the axis of the double source, is 1/frb 4 . That the ratio 

 L. 18 



