60 PEOBLEMS IN THE CONDUCTION OF HEAT " [35
la like manner we may treat a constant source distributed over a spher If the rate per nnit time and per unit of area of surface be Sn, we fin as above, for inside the sphere (c) .
and outside the sphere
2n +
and these forms are applicable to any integral n, zero included. Comparir with (44), we see that
which does not differ from (50), if in the latter we suppose n = integer -f- $.
The solution for a time-periodic simple point-source has already bet quoted from Kelvin (IV.). Though derivable as a particular case from (<• it is more readily obtained from the differential equation (1) taking here t] form — see (38) with n — 0 —
d* (rv) __ d? (rv)
~~dt~ dr^'
or if ?; is assumed proportional to eipt,
d* (rv)jdrz - ip (rv) = 0, ......................... (54)
giving rv — Aeiyt e"*-^, .............................. (55)
as the symbolical solution applicable to a source situated at r = 0. Denotii by q the magnitude of the source, as in (5), we get to determine A,
dr ,.=0
so that v — JL. #vt e-$&r ........................... (56)
4?rr
If from (56) we discard the imaginary part, we have
corresponding to the source q cos pt.
From (56) it is possible to build up by integration solutions relating various distributions of periodic sources over lines or surfaces, but an ind pendent treatment is usually simpler. We will, however, write down t integral corresponding to a uniform linear source coincident with the a2 of 2. If p2 = a? -{- yz} r2 = #2 + p2, and (p being constant) rdr = z dz. Th putting in (56) q — q1 dz, we get
eipt r°° e-rV(*P)rfr
(58)se solutions should coincide with the value obtained from (33) by putting cr = a-' dt and integrating with respect to t from 0 to oo . Or