APPENDIX: ANALYSIS 



The analysis follows somewhat that in Sommerfeld.* 



The diffusion equation in one dimension is 



d 2 u 1 du , . 



dz 2 = k dt 



where u is the concentration of solute, t is time, and k is the diffusion coefficient. 

 Assume that the variables are separable; that is 



u = Z(z) T(t) (2) 



Use — ( — 1 as the separation coefficient (where 2/ is the length of the specimen). 

 Then: 



Z» + (™J Z = 0, T'+ (™)\%T=0 (3) 



The solutions become 



„ /ixnz\ . hnz\ _, -It)* kt 



Z n = a n cos I 1+ 6„sin (- — I , T = e \ I 



A general solution is obtained from the superposition of a number of such 

 solutions for a proper set of values of n. 



Case of the Ring 



Take z = at some point on the ring, and assume that the initial concen- 

 tration f(z) is arbitrary but symmetrical about 2 = 0. The Fourier expression** 

 for f(z) becomes 



(4) 



f(z) = ^ A„cos (ip), A = 1 \ l x Hz) dz, An-jJ^ f(z) cos {^-\dz (5) 



The solution for u becomes 



u(z,t) = ^ A n e~\n fet cos« (6) 



Now consider f(z) to be a "6 function" situated at z = +£. The expression 

 for f becomes the theta function.*** Thus 



*Sommerfeld, A., Partial Differential Equations in Physi cs, Academic Press, 1949 

 **The sine terms drop out because of symmetry. 

 ***It is usually given for § = and for I = 1. 



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