52 PROBLEMS IN THE CONDUCTION OF HEAT [3
III. Continued point-source ; rate per unit of time at time t, an arbitrc function, / (t} :
IV. Time-periodic simple point-source, rate per unit of time at time q sin 2nt :
nk.r] ...................... (5)
Verify that v satisfies (1) ; also that — 4;7rrzdv/dr — q sin 2nt, where r = 0.
V. Instantaneous spherical surface-source ; a quantity Q suddenly gen ated over a spherical surface of radius a, and left to diffuse outwards a inwards :
To prove this most easily, verify that it satisfies (1) ; and further verify tb
,00
4?r vr2dr = Q; Jo
and that v = 0 when t = 0, unless also r — a. Remark that (6) becon identical with (2) when a = 0 ; remark further that (6) is obtainable from by integration over the spherical surface.
VI. Constant spherical surface-source; rate per unit of time for t whole surface, q :
[f"3 (,— (r—a) */4t __ - (r+a) 2 -«./.*
1 art1/2 = q/4i7rr (r > a) = q/'4s7ra (r < a).
The formula within the brackets shows how this obvious solution is c rivable from (6).
VII. Fourier's "Linear Motion of Heat"; instantaneous plane-souri quantity per unit surface, cr :
tre-*^
Verify that this satisfies (1) for the case of v independent of y and zt a that
*+co
v dx = cr.
— 00
Eemark that (7) is obtainable from (6) by putting Q/4nra? — a, and a = oo; directly from (2)'by integration over the plane.pposed equal to 2. may omit the factors already accounted for in (10). Expressions (7), (9) are of the standard form if we take