D,10 • GENERAL RESULTS 



\ I I 



X = X r= da 



X = — di 



Fig. D,10. 



where the eigenvalue differential equation for (p{x) is 



A; ^ + pcX'^ip = 



dx 



ki 



k = ki pc = piCi K = Ki = -— '- ip = <pi(x) 



p\Ci 

 k = k^ pc = P2C2 K = K2 = (f = <P2ix) 



P2C2 



with exterior boundary conditions 



d<pi 



ki -r^ = h<pi X = —d\ 

 dx 



d(p2 

 dx 



and interior boundary conditions 



(fl = (p2 



X = 



k,^ = k2^ x = 



dx dx 



(10-1) 

 -di ^ X ^0 

 ^ X ^ d2 



(10-2) 

 (10-3) 



(10-4) 

 (10-5) 



The eigenfunction obtained from Eq. 10-1 is 



<p{x) = { 



, . \x , ., . \x 



<P2{x) = a2 cos —= -\- 62 sm — -= 



VK2 VK2 



(P\{x) = ai cos — -^ + 61 sin — ^ —diSx^O 



^ a; ^ ^2 



(10-6) 



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