RECTANGULAR W AVE GUI DOS 143 



(vS-22) for a more precise statement of the assumptions) and the dots denote 

 (lifTerentiation with respect to v. We shall reserve primes to denote trans- 

 l)osition of matrices. It is supposed that Amn = A^m (equations(2-4) 

 plus (2-5) satisfy this condition and (1-4) plus (1-5) may be made to do so 

 by setting Fo = 2'/-7''o). 



The solution of (3-18) is approximately 



.V 



y,n = Z S,n,[e'un + e-^^/|] (3-19) 



(=\ 



where the d( are the 2N constants of integration and 



N 

 n=l 



^tjls\(=\ (3-20) 



n=l 



^( = ipc dv 



serve to determine ^( , ^t , and S,n( (the last to within a plus or minus sign). 

 We assume the .V roots (^J , (fo , • • • <^^ of the determinantal equation arising 

 from the lirst of equations (3-20) to be unequal, and denote by (pc that square 

 root of <^/ which has a positive real part or, if the real part be zero, which has 

 a positive imaginary part, v^f is any convenient constant. 



The approximation (3-19) may be obtained by setting the assumed form 



y,n = gm e^\ ^ = I <P dv 



in (3-18). The result is a set of N equations of which the wth is 



g,n ± 2g,„<p ± g„,>p + gni<p' ^ X) Amngn. (3-21) 



n 



We also assume 



I ^ I « k" M ^- I « I i-V' I « 1 gn.<p' 1 (3-22) 



gm = gmO + gva + gm'l + " ' " 



where gmr and its lirst two derivatives satisfy inequalities of the type 



1 gmO 1 » I gva I » I gm2 I • • • 



The first and second order terms in (3-21) give, respectively, 



gmO^~ — 2-( AmngnO = H- ? U 



