248 



THEORETICAL RESULTS ON NONSTANDARD PROPAGATION 



particularly helpful in this case. 

 Let 



y(h) = 2/(0) + b x h + bji? + - - • +b n h n + ■■■, 



I dh IV a^ ! V ~ 2& 2 



* " ^ (A = 0) = h ' h = ~br> 



■" _ (-66,fti + 12fr 2 2 ) 

 6x« 



. (120bi6 2 & 3 - 246! 2 & 4 - 1206, 3 ) 



« = r- = . etc. 



Ol 7 



9 = elr "(^)*.w = A m + y(0) 



(6) 



then 



A m = -2/(/u) = -2/(0) + 0M ( ^ I (« 2 ) 



k {hiY ~ 165 (/!s) 



+ 



e8 rzL( flo ) 3 __33_ 



L375 V2J 175 



(ft 2 ) (fts) + ^(ft 4 ) 

 3o 



ftl = 



— hw + 2? w 2 



3! 



w 3 +■ 



+ •••,(7) 



(8) 



where 



«2 



ft . ft 



As a further check, we treated the case a = +20, 

 X = 0.6356, for which Pearcey and Whitehead 156 

 give a value Ai = -10.21 + 1.07 X 10" 13 i. Equa- 

 tion (7) yields A x = —10.22, while the imaginary 

 part obtainable from Gamow's formula is 1.24 X 

 10- 13 i. 



It must be emphasized that the value obtained 

 from equation (7) should be verified by carrying 



out the integration of F(hi) = J VV(ft) + A m dh. 

 While doing so, one may as well compute 



Hrr-ty (/li) 



dh 



(10) 



Equations (7) and (8) are of the nature of asymp- 

 totic formulas; they should be terminated when the 

 individual terms begin to increase, and the error in 

 A m or ftx is then of the order of magnitude of the 

 last term retained. 



The following examples in Table 1 illustrate the 

 degree of accuracy obtainable from equation (7). 



Table 1. Approximate determination of A! from equation 



Vy(h) - 2/(ft0 

 and then obtain a correction to ft x by Newton's 

 method. 



The method of solving equation (3) explained 

 above has been found especially useful in the treat- 

 ment of substandard refraction, and to a lesser extent 

 in the treatment of the trapped modes in case of a 

 surface duct. In the latter case one can, of course, 

 solve for A m directly by computing F(hi) by numerical 

 integration. The method is not applicable for the 

 leaky modes in case of a surface duct. 



So far the discussion has centered on the solution 

 of equation (3), which in itself is only an approximate 

 asymptotic formula valid for large values of k. Let 

 the value of A m which satisfies equation (3) be 

 denoted by A m (0) ; then an improved value for A m 

 can be obtained from 



(11) 



where 



dh 



Vy (ft) + A„ 



dy 

 dhl' 



etc. , (12) 



and the derivatives of y are to be evaluated at h = hi. 

 (7) and the verification that ] \/y(h) + Ai dh = 2.383.* 



* y(h) = h + ce- Xh . 



