436 Lord Rayleigh on the Light emitted from a 



If the question of the phases of the two sources be left- 

 open, (19) gives 



2 + 2 cos (^-^EH*? (20) 



If D be small, this reduces to 



2 + 2 cos (e 1 -e 2 ), 



which is zero if the sources be in opposite phases, and is 

 equal to 4 if the phases be the same. 



If in (20) the phases are 90° apart, the cosine vanishes.. 

 The work done is then simply the double of what would be 

 done by either source acting alone, and this whatever the 

 distance D may be. If this conclusion appear paradoxical, 

 it may be illustrated by considering the case where D is 

 very small. Then 



— 47rK <£=cos>(_p£ + € — £R )+ cos (pt + e + ^ir — kRo) 



= \/2 .C0s(pt + €±%7T — E ), 



representing a single source of strength \/2, giving intensity 

 2 simply. 



We have seen that the effect of a number n of ui it 

 sources depends upon the initial phases and the spatial dis- 

 tribution, and this not merely in a specified direction, but in 

 the mean of all directions, representing the work done. We 

 have now to consider what happens when the initial phases 

 are at random, or when the spatial distribution is at random 

 within a limited region. Obviously we cannot say what the 

 effect will be in any particular case. But we may inquire 

 what is the expectation of intensity, that is the mean intensity 

 in a great number of separate trials, in each of which there 

 is an independent random distribution. 



The question is simplest when the individual initial phases 

 are at «random in separate trials, and the result is then the 

 same whether the spatial distribution be at random or pre- 

 scribed. For the mean value of every single term under the 

 sign of summation in (19) is then zero, D meanwhile being 

 constant for a given pair of sources, while 



cos^ — e 2 )_=0. 



9 



t/0 - 



The mean intensity, whether reckoned in all directions, or 

 even in a specified direction (16), reduces to n simply. 



If the sources are all in the same phase, or even if each 

 individual source retains its phase, cosfe — e 2 ) in (19) 



