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Full text of "Scientific Papers - Vi"

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
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        (, (ra) */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 :
Verify that this satisfies (1) for the case of v independent of y and zt a that
v dx = cr.
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