300 Lord Rayleigh on the Production 



in the same phase be arranged in a vertical line so that th«- 

 distance between immediate neighbours is W the work 

 done by each is the same as if the others did not exist. 

 The whole work accordingly is ?i, whereas the work to be 

 done at a single source o£ magnitude n would be n-. Thus 

 if the sound be only wanted in the horizontal plane, the 

 distribution into n parts effects an economy in the proportion 

 ofn:l. It is not necessary that all the possible places be- 

 tween the outer limits be actually occupied. All that is 

 necessary is that there be n equal sources altogether, and 

 that the distance between any pair of them be a multiple 

 of i\. 



Returning for the moment to the case of two source- 

 only, we may be interested to estimate the work consumed 

 by following the law of emission of sound to a distance in 

 the various directions. If //- be the cosine of the angle 

 between any direction and the vertical, the relative retarda- 

 tion due to the difference of situation is //-D. If the potential 

 at any great distance due to one source is cos to*, that due 

 to the other may be represented by cos^(af— /xD). For the 

 aggregate potential we have 



cos kat(l 4- cos fikD) -f- sin kat sin /x^D, 



or for the intensity 



2(l + cos/x^D) (12) 



This is in direction /x. For the total intensity over angular 

 space we must integrate with respect to fi from —1 to +1. 

 The mean intensity is thus 



2f'(l + co3MD)rf^=2(l+^^). . . (13) 



The scale of measurement is at once recovered by supposing 

 = 0, in which case the intensity in various directions would 

 be uniform. The ratio of the mean intensities, which is also 

 that of the work done, is thus 



' 4(1-^-^) (1*) 



This is the ratio in which the work done is diminished when 

 a source is divided into two parts and these parts separated 

 to a distance D. 



While from the theoretical point of view there is no doubt 



as to the saving that might arise from the use of a number 



of separated sources, it is to be noticed that the saving is in 



the pressure. Since at the present time most of the pressure 



* It is not necessary to exhibit the dependence on ;•. 



