186 TECHNICAL SURVEY 
at h— ©, U, should represent an upgoing wave 
only. Here h denotes height in feet. 
y(h) = Nh) —-1=2 X 10° M(h) ,k = 2 (9) 
and A,, is the characteristic value which is generally 
complex. It is convenient to introduce natural units 
of height 
ou h — ()-27)-3 LGN —9 —1 
2= 7H = (a)4,g = Gp = 2:36 X 10% cm™, 
4 
whereby equation (8) is transformed into 
— oP E + f(z) + Da | CG) =O. Cul) 
The term f(z) in equation (11) represents the refrac- 
tion anomaly and is equal to zero for a standard 
atmosphere. In the first instance we shall be treating 
the case where 
f(z) = ae, (12)¢ 
and we shall later generalize the treatment to deal 
with any M curve represented as a series of Laguerre 
functions. If the original M curve is represented by 
the expression 
M(h) = bh + ae , b = 0.036 ft, (18) 
then a and ) are obtained as follows: 
4 
a= 2x 10*(E)'a,n = cH. (14) 
It is to be noted that in contrast to the constants a 
and c in equation (13), which are independent of 
frequency, the constants a and \ in equation (12) 
are frequency dependent. For a given observed M 
curve the constants a and A will therefore differ with 
the frequency band used, as will also the neight 
represented by one unit of z. 
FORMAL SOLUTION OF THE PROBLEM 
BY THE PERTURBATION METHOD 
In order to solve the equation 
aU (2) 
dz* 
i E Taps D, | TG) 0, CS) 
we seek a solution in the form 
CJ 
U;, = yy AimU n° (2) ? (16) 
m=1 
where U,,°(z) are the height-gain functions of the 
m-th mode in the standard case, which satisfy the 
“No confusion should arise from the use of \ in equation 
(12) and the standard usage of \ to denote wavelength. 
equation 
@U7°(2) 
dz? 
i [: i Da! | Ut) — 0) a) 
i [ U,,°(2) } dz=1. (18) 
The solutions of equations (17) and (18) are: 
Uni(2) = Cnut Hy® (u) , w= 2 (2 + Dy®)$ ,(19) 
1 
can for SQ) [2100 ra. 
1D) = Tres - m= (3) ) (21) 
2 
where 
Tx (Um) + J_4 (Um) = 0. (22) 
For small z the power series development of U,,°(z) 
is useful: 
Une) = 1) Alz* (23) 
py i 
1 . 
Ap= — pga [Pettus Ae], 20 
DE\e | DO 
(Goal cee a 
while for large z one may use asymptotic expansion 
of equation (19) 
(25) 
Hy®(u) > ee ei(— u + 5m/12) . 
TU 
bi 385 
[2 + 72u 10,368u2 * | (26) 
If now the expansion (16) be substituted into 
equation (15), we obtain, on making use of equation 
(17), the condition 
> Arm | — D,°) + ae 7 U,(2) = 0. (27) 
On multiplying this equation by U,°(z), where n is 
any integer, and integrating from 0 to © we get a 
system of equations for the determination of D, and 
the A pm: 
yD Aim [ sal D,,°) 8nm + 2B on(d) | =0, 
ii n=1,2,3--- es) 
C-} 
Bnm(A) = [ Un(z) Un(2) edz. (29) 
