D,10 • GENERAL RESULTS 



By the results of Art. 2 we determine the ampHtudes An upon expansion 

 of Q{x, 0) = —1 in terms of the eigenf unctions <pn{x). Thus we obtain 

 from Eq. 2-7 



/ pCifndX PlClcpindx + / P2C2<P2ndx 



A-n — 



j pc^\dx J_^^ piCKpl^dx + f ' p2C2<plndx 



At this point it is convenient to introduce the following nondimen- 

 sional quantities: 



di di d\ dl 



iAnTl = 



2 .2 //^2n di \kx\ 



Then the characteristic equation for jui (or /X2) is 



1 tan Ml + Ml . ^1 ii2 ,r Ml /'iA'7„\ 



tan iJ-2 = 1- J-> Ni = -J- (10-7a) 



1 — /ii tan jLti K2\Ki K\ 



and the expression for the amplitude in Eq. 10-6 becomes 



plClHln -f- P2C2n.2n 



where 



Gin = — [sin llln + hln (cos Mln " 1)] 

 Mln 



J 



(72n = — [sin ll2n " 62n (cOS M2n " 1)] 



M27I. 



^In = TT^ [(1 + &lrt)Mln + (1 " &ln) slu /Xl„ COS /il„ - 2&in Slu^ /ii„] 

 ^Mln 



H2n = ?r^ [(1 + 61 jM2n + (1 " &2n) slu JLi2ri COS JLt2n + 2&2n slu^ /Li2n] 

 ^M2n 



Thus we obtain for the temperature distribution: 

 T{x, t) 



= e(^, r) 



cos Min^i + bin sin Min^Oe""'"''' — 1 ^ 6 ^ 0" 



(10-11) 



1 + 2j ^" (cos /X2n^2 + &2n sin M2n?2)e-''L^^ ^ ^2 ^ 1 



where the pun's iti2n's) are computed from the eigenvalue equation (Eq. 

 10-7a), the amplitudes An are determined by Eq. 10-10, and the 6„'s 

 are computed from Eq. 10-8 for ain = 1. The first two eigenvalues satisfy- 

 ing Eq. 10-7a are obtainable from curves in [18]. 



(275) 



