608 Prof. H. S. Carslaw on Bromioielis Method of 



5. Rod of length b composed of two different materials. The 

 ends x — Q and x = b kept at zero and v respectively. 

 Ihe initial temperature zero. 



Let v 1 be the temperature in the first part of the rod 

 (0<#<a), and K 1? c l5 pi be its conductivity, specific heat, 

 and density. Let v 2 be the temperature in the other part 

 (a<x<b), and K 2 , c 2 , p 2 the corresponding constants therein. 



k x = Kx/c^! and k 2 = K 2 /c 2 p 2 . 



The equations for the temperature are as follows : — 



(2) ^1=0, when# = 0: *'a— v o> when x — b. (2') 



(3) w'i=0, when£=0: v 2 =0, when £ = 0. (3') 



v 1 =v 2i when x = a (4) 



Kl ^=E 2 |^, when* = a (5) 



<$x ^x 7 



It is clear that 



v 1 = A 1 smaxe~ KlCt2 \ 



an d v 2 = (A 2 sin /*«(#— a)+B 2 sin/ia(6~ x))e~ K ^\ 



satisfy (1) and (1'), when fx= V^iAs)- 



They also satisfy (4) and (5), provided that 



A 1 sin eta = B 2 sin //,« (6 — a) , 



and Ki Ai cos aa = K 2 /a( A 2 — B 2 cos ^,a(6 — a)). 



Therefore we take 



A 2 = (cr cos act + sin aa cot /ita(& — a)) A 1? 



and B2= sin / x a (6-a) Al ' 



k; 2A 6 v \K 2 c 2 /? 2 /* 



Introducing the path (P) and choosing a suitable value for 

 A l5 we are led to the solution : — 



?; Tsui ax e~ Kia ^ , /£ > s 



„ =r -°l_— — da (6; 



1 nrj _b (a) a 



_ Itt f/sin yLta (> — a) sin aasin//,a(& — #) \ g^" 2 * , ,^ 

 V2 ~ V J\smfi*(b-aj F(a) sin^«(6— a)/a 



where <t= — 



