Random Distribution of Luminous Sources. 433 



centres, it fails to satisfy strictly the requirement that all the 

 centres act independently, for some of them will lie at dis- 

 tances from nearest neighbours less than the number of wave- 

 lengths necessary for approximate independence. The simple 

 conditions just discussed are thus an ideal, approached only 

 when the spacing is very open. 



We have now to consider how the question is affected 

 when we abandon the restriction that the spacing of the unit 

 centres is very open. The work to be done at each centre 

 then depends not only upon the pressure due to itself but also 

 upon that due to not too distant neighbours. Beginning 

 with a single source, we may take as the velocity-potential 



cos k(at-r) 



where a is the velocity of propagation, k — 2irl\ and r is the 

 distance from the centre. The rate of passage of fluid across 

 the sphere of radius r is 



4:7rr 2 d<f>/dr = cos Je(at— r) — kr sin k(at — r). . (10) 



If Bp denote the variable part of the pressure at the same 

 time and place, and p be the density, 



~ d<b oka sink (at — 7") ,. „ v 



S P=-Pdi=- 4^ • • * < U > 



The rate at which work (W) has to be done is given by 



d W __ rs , 3 d(f> __ pka sin k(at — r) 

 dt " ' dr ±7rr 



x \_kr sin k(at — r) — cos k(at— r)], . . (12) 



of which the mean value depends upon the first term only. 

 In the long ran 



W/t=pk 2 al8>rr (13) 



It is to be observed that although the pressure is infinite at 

 the source, the work done there is nevertheless finite on 

 account of the pressure being in quadrature with the prin- 

 cipal part of the rate of total flow expressed in (10). 



When there are two unit sources distant D from one 

 another and in the same initial phase, the potentials may be 

 taken to be 



cos k(at — r) cos k(at-r') , 1X 



*= — ^Tr — > * = w — • • (11) 



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