1918]
RANDOM DISTRIBUTION OF LUMINOUS SOURCES
571
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
representing a single source of strength \/2, giving intensity 2 simply.
We have seen that the effect of a number n of unit sources depends upon the initial phases and the spatial distribution, 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 prescribed. 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
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, cos^ — e2) in (19) remains constant in the various 'trials for each pair, and we have to deal with the mean value of sin kD 4- kD when the spatial distribution is at random. We may begin by supposing two sources constrained to lie upon a straight line of limited length I, where, however, I includes a very large number of wave-lengths (X).
If the first source occupies a position sufficiently remote from the ends of the line, so that the two parts on either side (4 and £2) are large multiples of X, the mean required, represented by
smkDdD Ijf^sinkDdD o kD I, +/J0 kD" 12
.(21)
may be identified with ir/kl, since both upper limits may be treated as infinite. Moreover, nrjkl may be regarded as evanescent, kl being by supposition a large quantity. write