D • CONDUCTION OF HEAT 



Substituting the expression for ^ in the four boundary conditions, we get 

 four homogeneous Hnear algebraic equations for the constants of inte- 

 gration ai, a2, 61, and h^: 





—7= ) - 61 1 1 H — TV" ^^'^ ^~r 



«i Vv^ - tan ^^ - 61 1 + Vt^ tan -^ = (10-2a) 



a2 tan —= — 62 = (10-3a) 



ai — a2 = (10-4a) 



foi -^ - 62 ^ = (10-5a) 



V/Cl V«2 



These equations possess nontrivial solutions if the eigenvalue X is a root 

 of the characteristic equation: 



^Han^ + 1 



J^ ^ tan ^ - 1-^ .P = (10-7) 



—j-^ tan —^ 



This characteristic equation yields, of course, an infinite set of eigenvalues 



X =-. X„ (n = 1, 2, . . .) 



For each eigenvalue X„ there exists a set of integration constants (ain, azn, 

 hin, ?>2n) and the ratios of any three to the fourth one can be determined 

 from the boundary conditions, Eq. 10-2a through 10-5a. Thus 



Clin -, Oln ki \k\ , \ndl 02n , X„a2 



, — = Y- \\— tan — T=j — = tan — -= (10-8) 



ftln flln r^X \ K-1 '\/k2 '^^'^ 'VK2 



The particular solution corresponding to X„ is 



[Oln = ^\n{x)4/niS) = ( COS — ^ + 61 slu — 7= ) g"''"' 



e„ = ^„(a;)^„(0 = <^ ^ ^^ Xa:\ . 



62n = <p2n(a;)i/'„(0 = (cos -^ + 61 sin -^) e~^»' 



(10-9) 



where, without loss of generality we have set ai„ = 1, since this constant 

 can always be included in the amplitudes An which appear in the general 

 solution (cf. Eq. 2-6), 



e(a:, t) =^AnQn = ^ An^n{x)e-^'r^^ 



T{x, t) 



T, 



= e{x, = 1 + Q{x, t) 



<274) 



