COMPUTATIONAL PROGRAM FOR THE EXPONENTIAL MODEL 



255 



tion (2), since the coefficients A km will be obtained 

 simultaneously with the D k when the iteration pro- 

 cedure is used. This will be possible only in a limited 

 region of low altitudes, since at great heights the 

 U m °(z) increase rapidly in magnitude as m is 

 increased. However, near the ground the U m °(z) are 

 all of the same order of magnitude ( = iz) and 



= s° 



dC k (0) 

 dz 



-*s 



(69) 



V = 



z = 



II = 



X = 



L = 



h a = 



D m 

 X 



If this derivative of U k (z) at the ground can be 

 obtained with sufficient accuracy, then one may use 

 it to integrate numerically the original equation (11). 

 It is well known that, for a given order of the deter- 

 minant used, the characteristic values D k are 

 obtained with higher accuracy than the height-gain 

 functions. 



It may be added here that /?n(X) computed from 

 equation (44) agrees up to X = 5.0 with the values 

 given by Pearcey and Tomlin. 106 



The perturbation method will of course become 

 inefficient when trapping conditions are approached. 

 For such values of a and X, asymptotic methods may 

 provide approximate values for the D k , provided 

 care is taken at each stage to estimate the order of 

 magnitude of the error involved. It is planned to 

 map out by a combination of these methods the 

 real and imaginary parts of D k in the operationally 

 relevant region of the a, X plane. 



Symbols for Use in 

 Theory of Nonstandard Propagation 



q = standard slope of N 2 curve = 2.38 • 10 -7 m _1 . 



p = slope of lower section of N 2 curve in bilinear 



model . d 



^ = e 



JV 2 - 1 = 2M • 10- 6 . 



(fc 2 g)S /[ = h/II height in natural units 

 (fc = 2tt/X) . 



(k 2 q)~* = 7.24 X C J (feet) natural unit of height . 



1/2 (kq-)' d = d/L distance in natural units . 



2 (kq 2 )~* = 6.69 X cm * (thousands of yards) = 

 natural unit of distance . 



anomaly height (height of joint in bilinear 

 model) . 



h a /H anomaly height in natural units . 



characteristic value (for y = at h = h a ) . 



(fc/g)' A m = B m + iA m characteristic value in 

 natural units . 



s -2 D (abbreviation for use in computing) . 



etoit — 21TM/X — i n /i 2^5 



plane wave 



L 



depends 

 on X 



OD 



-* 2 e- Am * + iB ™ x ■ U m ( Zl ) U m (s 2 ) 



1 

 i 



natural units only 



OD 



U 2 dz = 1 . 



slant range . 

 horizontal range 



