1911] PROBLEMS IN THE CONDUCTION OP HEAT 61
In considering the effect of periodic sources distributed over a plane xy, we may suppose
v oc cos Ix . cos my, ........................... (59)
or again v oc Jn (Icr) . cos nd, ........................... (60)
where r2 = x* + yz. In either case if we write I2 + in" = k2, and assume v proportional to eipt, (1) gives
dzv/dzz = (k* + ip)v ............................ (61)
Thus, if
Jc* + ip =.R (cosa + isin a), ....................... (62)
where A includes the factors (59) or (60). If the value of v be given on the plane z = 0, that of A follows at once. If the magnitude of the source be given, A is to be found from the value of dvjdz when z = 0.
The simplest case is of course that where k = 0. If Veipt be the value of v when z = 0, we find
v= Fete*e-Wi»p) ; ............ -. ............... (64)
or when realized
Vss Ve-Wa»#cos{p*-*V(.p/2)}> ................... (65)
corresponding to
v= V cos pt when z = 0.
From (64) - f — J = *J(ip). 7e*« = \ae[^, ..................(66)
if cr be the source per unit of area of the plane regarded as operative in a medium indefinitely extended-in both directions. Thus in terms of o-,
v = ^~*i(^e-^^,.........................(67)
or in real form
..............(68)
•" Yjf
corresponding to the uniform source a cos pt
In the above formulae z is supposed to be positive. On the other side of the source, where z itself is negative, the signs must be changed so that the terms containing z may remain negative in character.
When periodic sources are distributed over the surface of a sphere (radius = c), we may suppose that v is proportional to the spherical surface harmonic Sn. As a function of r and t, v is then subject to (38); and when we introduce the further supposition that as dependent on t, v is proportional
to eipt, we have
d*(rv) n(n
dr* r2
(rv) — ip (rv) = 0................(69)) independently. ndv\ . 1 d*v . , . ,^ rt