574 ON THE LIGHT EMITTED FROM A [486
This potential is next to be multiplied by 4f7rRzdR and integrated from 0 toE. We find
(27)
We have now to divide by F2, or lQ7r"RG/9 ; and finally we get
(19) = n + -^^(sin kR - kR cos kR}\ ............ (28)*
where kR will now be regarded as very large. When n is moderate, or at any rate does not exceed k3R3, the second term is relatively negligible, that is reduction occurs to n simply, provided n be not higher than of order R3/~\?, corresponding to one source for each cubic wave-length f. But evidently 71 may be so great that this reduction fails, unless otherwise justified by a random distribution of initial phases.
At the other extreme of an altogether preponderant n, the second term in (19) dominates the first, and we get in the case of constant initial phases and a very large kR,
Under the suppositions hitherto made of a random spatial distribution within the sphere (R), and of uniformity of initial phases, there is no escape from the conclusion that the reduction to the simple value n fails when n is great enough. Nevertheless, there is a sense in which the reduction may take place, and the point is of importance, especially in the application to the dispersal of primary waves by a cloud of small obstacles. In order better to understand the significance of the term in n2, let us calculate the intensity due to an absolutely uniform distribution of source of total amount n over the spherical volume. Since there is complete symmetry, it suffices to consider a single specified direction which we take as axis of 2. As in (15), we have
nei(pt-lcR0) rrr
~ 4i-rrR0<l> = - p - Ml eik*dasdydz, ............... (30)
as the symbolical expression for the velocity-potential, from which finally the imaginary part is to be rejected. The integral over the sphere is easily evaluated, either as it stands, or with introduction of polar coordinates (r, 6, «u) which will afterwards be required. Thus with /* written for cos 6,
R r+i
eikz dxdydz-^Tr] eikr* r2 dr dp,
o J -i R
4vr rR _ 4,^
= -y- sin kr . rdr -j^ (sin kR kR cos kR) ....... (31)
K J o AT
* We may confirm (28) by supposing UR very small, when the right-hand member reduces to n2.
f The number of molecules per cubic wave-length in a gas under standard conditions is of the ' order of a million.stribution of Sound," Phil. Mag. Vol. vi. p. 289 (1903); Scientific Papers, Vol. v. p. 136.are aiming