190 TECHNICAL SURVEY 
By successive differentiation of equation (43), it 
is possible to express any high order derivative of 
Bam(A) in terms Of Bam(X). From a purely formal point 
of view we can say therefore that by our method we 
can treat any M anomaly by expanding it into a 
series of Laguerre functions, since these functions 
involve only terms of the form z*e~”. It may be 
pointed out that a single term z*e~” vanishes both 
at the ground and at great height and reaches a 
maximum at z = k/d. Such a single term is therefore 
suitable to represent an elevated duct. 
COMPUTATIONAL PROGRAM 
FOR THE EXPONENTIAL MODEL 
The Analysis, Section of the Columbia University 
Wave Propagation Group has undertaken the com- 
putation of B,n(A) for \ = 0(0.1)4.0 and n,m = 
1, 2, 3, 4, 5. With these functions tabulated, it is 
planned to compute the characteristic values D,, for 
such values of a and ) that the difference between the 
values of D, obtained from the fourth order deter- 
minant and from the fifth order determinant will be 
only about 0.01. The program also calls for the 
computation of the height-gain functions from equa- 
tion (2), since the coefficients A,m will be obtained 
simultaneously with the D, when the iteration pro- 
cedure is used. This will be possible only in a limited 
region of low altitudes, since at great heights the 
Un (2) increase rapidly in magnitude as m is 
increased. However, near the ground the U,,°(z) are 
all of the same order of magnitude (= 7z) and 
oy Aree (69) 
m=1 
ae dUx (0) _ 
If this derivative of U,(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, are 
obtained with higher accuracy than the height-gain 
functions. 
It may be added here that Bu(A) computed from 
equation (44) agrees up to A = 5.0 with the values 
given by Pearcey and Tomlin.1% 
The perturbation method will of course become 
inefficient when trapping conditions are approached. 
For such values of a and A, asymptotic methods may 
provide approximate values for the D,, 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, in the operationally 
relevant region of the a, ) plane. 
Symbols for Use in 
Theory of Nonstandard Propagation 
= standard slope of N? curve = 2.38-10~ m~! 
= slope of lower section of N? curve in bilinear 
model . 
=s3. 
= N?—1=2M - 10- 
= (kg)! h = h/H height in natural units 
(kK = 2/d) . 
H = (k?q)+ = 7.24 don! (feet) natural unit of height . 
x = 1/2 (kq?)! d = d/L distance in natural units . 
L = 2 (kq?)+ = 6.69 Aun? (thousands of yards) = 
natural unit of distance . 
xan © KIS BK 
h, = anomaly height (height of joint in bilinear 
model) . 
g = h,/H anomaly height in natural units . 
An = characteristic value (for y = Oath = h,). 
Dm = (k/q)! Am = Bn + i1Am characteristic value in 
natural units 
X = s-? D (abbreviation for use in computing) . 
etwt — 2mid/\ — ix/4 Ont 
ony 
Ve 
depends 
on A 
W=e 
plane wave 
aa eam? + iB, . Un (21) Un (22) 
» 
ee 
natural units only 
[urea 
0 
R = slant range . 
d = horizontal range . 
