382 Lord Rayleigh : Problems in 



IV. Time-periodic simple point-source, rate per unit o£ 

 time at time t, q sin 2ni : — 



v = -r— e- Vtt.f sin \2nt — 7i* . r~|. . . . (5) 

 \nrr L J 



Verify that v satisfies (1) ; also that — ±7rr 2 dv jdr = q sin 2 nt 

 where r = 0. 



V. Instantaneous spherical surface-source ; a quantity Q 

 suddenly generated over a spherical surface of radius a, and 

 left to diffuse outwards and inwards :-— 



To prove this most easil} r , verify that it satisfies (I); and 

 further verify that 



4:7r I vr 2 dr = Q; 



Jo 



and that v = when t = 0, unless also r = a. Remark that 

 (6) becomes identical with (2) when a = ; remark further 

 that (6) is obtainable from (2) by integration over the 

 spherical surface. 



VI. Constant spherical surface-source; rate per unit of 

 time for the whole surface, q : 



— q\\.irr (r>a) =q/±7ra (r<a). 



The formula within the brackets shows how this obvious 

 solution is derivable from (6). 



VII. Fourier's " Linear Motion of Heat"; instantaneous 

 plane-source ; quantity per unit surface, a : — 



V ~ 27T 1 / 2 t 1 ! 2 ^ ' 



Verify that this satisfies (1) for the case of v independent 

 of ij and z, and that 



"+CO 



v dx=o-. 



r 



Remark that (7) is obtainable from (6) by putting Q/47ra 2 = <7, 

 and a = co ; or directly from (2) by integration over the 

 plane. 



