APPEEDIX F 



Derivations Relating "^ with > d , and ^ C with d d. 

 SX Tx BY 5y 



Given: 



Then: 



oT tanh f^PITdl 

 ^ |_GT J 



tanh 



Let: 



Then: 



Given 



2Ttd _ tanh ""'" f2Tlc| 

 CT ~ gfj 



d = CT . ^ . fin [l + 2Trc\ - In [l - 2TrC \ 

 211 |_ \ gT / I gT / 



k' = T/i|ir and k" = 2ir/gT 



d - Ck' Hn (1 + k"C) - In (l - k"C)] 



C = f(d) and d = g(X,Y) 

 However^ it may also be considered that: 



d = F(c) and C = G(X,Y) 



Therefore (Kaplan, 1952;, p. 86): 



"id _ d(d ) . >C and"^d ^ d(d) . •a_C 

 "JX dC ^X fj dC >Y 



■^d _ k' 

 71 



C d_[ln(l+k"C) - ln( 

 dc\ 



Similary: 



^ ^ k' |c/ k" 



^X 1+k^ 



^ =^ . 1 

 ^X 5X k' 



\C ^■^ . 1 

 SY ^Y k' 



l-k"C)Hc +/] 



(-k'')V c + /] 



l-k"C/dX I 



ln(l+k"C) - ln(l-k"Cj 



ln(H-k"C) - ln(l-k"C)\^c| 



— 





1 







Ck" 

 Ll+K"C 



+ 



Ck" 



l-k"C 



+ ln(l+k"C) . 



- ln(l-k"C) 



~ 



1 







Ck" 

 _l+k"C 



i C k" 

 * l-k"C 



+ ln(l+k"C} . 



- ln(l-k"C) 



te] 



59 



