Chapter 2] 
PERTURBATION THEORY FOR AN EXPONENTIAL M CURVE 
IN NONSTANDARD PROPAGATION: 
ABSTRACT 
i THIs chapter a perturbation method is developed 
for treating nonstandard propagation in the case 
when the deviation of the M curve from the standard 
(= the M anomaly) can be represented by a term 
ae-™, where z denotes height in natural units. The 
method is also applicable to other forms of the M 
anomaly which can be derived from an exponential 
term by differentiation with respect to \; in fact, in 
its region of convergence, it is formally applicable 
to the most general type of M curve, including 
elevated ducts. The region of practical convergence 
of the method ranges from standard down to cases 
where the decrement is a small fraction of the 
standard value. 
The procedure followed is to express the height- 
gain function U,(z) of the k-th mode in the non- 
standard case as a linear combination of the height- 
gain functions U,,°(z) of all the modes in the standard 
case. 
Use) = )) ArnUn°@) « (1) 
The execution of this plan hinges on the possibility 
of evaluating the quantities 
oO 
Bnm(A) =|) U,,°(2) U,,°(z) edz . (2) 
It is shown that Brn(A) satisfies the differential 
equation 
AB am 
dy 
1 
= DN + Bam(X) pS 
1 
| - ato (Dn? + Da!) ++ Zs (Dn 
—D,0)| , (3) 
whose solution is 
1 
= 0_— 
a? 
Bis 0 os 
2 Pn FD) +t a5 a 
A 
3 
_z O+p 0H -= 41 (py 0_p_ 02 
= es pyr Oy 
0)2 
D,) 
Bum () = 
r 
es 
oVz 
(4) 
Here D,,° denotes the characteristic value of the 
m-th mode in the standard case. For large » the 
"By C. L. Pekeris, Columbia University Wave Propagation 
Group. 
185 
following asymptotic formula holds 
Bam => 
2 
[>s+20 (D,.° + D,°) — 2+ (Dn - v| 
8 ES + 2d (Dn° + D,°) — : (Dn° — Dat)| 
3 
[» + 2d (Dn? + D,°) — 2 +; (DP= 40) | 
Having determined the B,,(A) from equation (4), 
or by a numerical solution of equation (3), the 
characteristic values D, and the coefficients A,,, are 
to be solved from the infinite system of equations 
» Ain [ =F D,,°) Onm =F OBnm 0) | = 
m=1 
n= 1,2,3 0 0.0 (6) 
For this purpose a simple iterative procedure has 
been developed, which has been found to be rapidly 
convergent. The A,,,, are normalized by the condition 
[ue (2) dz=1= yy Alig (7)° 
m=1 
One can also expand D, as a power scries in a 
D, = Di+aDi+eDi+---, 
2 
DO = = Bas D© =e TS at. 
(T™ Sells 
An alternative expression for D is given in equa- 
tion (65). 
INTRODUCTION 
In the theoretical treatment of nonstandard 
propagation by the method of normal modes, one is 
confronted with the task of solving the equation 
CUn 
dh? 
Il 
+ ke [wm oF An | U, = 0, (8) 
subject to the condition that U,,(0) = 0 and that 
inm = 1, 
fn = 0. 
n= m 
nm =m. 
The integral | UK @)dz diverges when taken along the 
real axis; it converges, however, and to the same limit, when 
the path is a radial line in the fourth quadrant of the z plane. 
In the sequel, whenever an integral is divergent it will be 
understood that the path is suitably modified. 
