TnVfEfrWlllill-aiiHjWMMHn 



To make the algebra minimal we choose a definite set of conditions 

 at x ■ and at x ■ I, namely the simple junction conditions that there 

 Is no change to the total displacement or to the slope at x ■ 0, x M I, 

 the tension in all three strings being the same. Thus we take 



y(O-) + 1 - y(0+) 



y(Jl-) - y(£+) 



y'(O-) + ik Q - y»((»-) 



y'(*-) - y'(*+) 



(7.15) 



If k + k, the set of 8 equations (7.3, 7.8, 7.13, 7.1«, 7.15) has a 

 o • 



finite solution provided 



sin kj* i 

 (excluding r esonance of the middle portion) and then 



(7.16) 



y(*> 



2ik k t 



k 2 -k 2 

 o i 



cosec(k I) 



2k 2 

 y(CH-) f 



k 2 -k 2 

 o i 



2ik k 



2-i cot k £ 



k 2 -k 2 ! 

 o i 



(7.17) 



etc. This of course completes the solution for these simple problems, 

 for now that all constants are known, Y (s) is known frou (7.7), 

 Y_(s) from (7.2) and Yj(s) from (7.12). 



A method which shows how a generalized W-H equation may be treated 

 ignores the detailed solution (7.17), and instead eliminates the unknown 

 constants from (7.7), (7.2), and (7.12) in just the same way that the 

 corresponding constants were eliminated in $2 to get a standard W-H 

 equation. Thus we write 



38 



