D,2 • MATHEMATICAL FORMULATION 



In general, the eigenvalue problem leads to an infinite set of eigenfunc- 

 tions (fn with a corresponding infinite set of eigenvalues X„ which are ob- 

 tained by satisfying the boundary conditions. 



2. The eigenfunctions ^„, when multiplied by \^, form an orthogo- 

 nal set. Since this is perhaps the most important property of the eigen- 

 functions we shall indicate briefly the proof of this important result: If 

 k, p, and c are continuous with continuous derivatives in the domain G, 

 we have by a generahzed Green's formula [8, Vol. 1, p. 239]: 



^^--<pj-$^ds (2-3) 



// [iPnL{iPm) — (pmL{(pn)]dxdy = / k iifn 



dn 



dn 



where ds is an element of arc on T, and 



Substituting the differential equation and boundary condition (Eq. 2-2) 

 into Eq. 2-3, we get 



Hence 



{K - K) II pc^m<Pndxdy = 1^ h{<pm'Pn - <Pn(Pm)ds = (2-4) 



G 



// pccpmiPndxdy = for m 9^ n, \m ^ Xn 



so that the functions 



^n = \^ 



<Pn 



(n = 1, 2, 



.) 



form an orthogonal set. 



It often happens that the thermal properties k, p, and c are continu- 

 ous in different subdomains Gi, G2, etc., but are discontinuous across the 

 boundaries Ti, r2, . . . separating the domains, 

 as is the case in composite rocket walls. In 

 this case the general Green's formula must be 

 applied to the subdomains Gi, G2, etc., sepa- 

 rately and the results added. Suppose for the 

 sake of simplicity that the total domain G is 

 thus divided into only two subdomains, Gi and 

 G2, separated by the boundary Fi (see Fig. D,2) • 

 Let the wth and nth eigenfunction in G be 



(fim, <pin in Gl 



<P2m, <P2n lu G2 



Then, applying Green's formula in both do- 

 mains separately and adding, we have 



(p = 



Fig. D,2. Division of the 

 domain G into the subdo- 

 mains Gl and G2 by the 

 boundary Fi. 



< 257 ) 



