Chapter 21 



THE SOLUTION OF THE PROPAGATION EQUATION 

 IN TERMS OF HANKEL FUNCTIONS" 



The calculation of the field strength in the atmos- 

 phere depends upon finding a solution of a wave 

 equation incorporating the propagation properties of 

 the atmosphere and satisfying the boundary condi- 

 tions at the surface of the earth and for large heights. 

 This chapter shows how the wave equation, for cer- 

 tain specified conditions, may be solved in terms of 

 Hankel functions. 



Let 



2 = height of receiver above earth's surface, 

 h, = height of transmitter, 

 d = great circle distance between source and 



receiver, 

 X = wavelength, fc = 2x/X, 

 / = frequency, co = 2irf, 

 M = modified index of refraction, 

 a = radius of the earth. 



Under the simplifying assumptions of horizontal 

 stratification, slight variation of refractive index in 

 a wavelength, smooth earth's surface, the plane earth 

 representation, and the use of the simplified boun- 

 dary condition M? = for z = 0, which eliminates 

 the polarization of the source, the field of a (dipole) 

 source is described by a scalar wave equation: 



A 2 * + fc 2 M 2 *=0, (1) 



plus appropriate boundary conditions. Separation of 

 equation (1) in cylindrical coordinates leads to the 

 formal expansion for the field of a dipole source : 



•* = eiwt\\ - i-wH^ (kd cos a,) U n (z) U„(h t ) , (2) 



n- 1 



where Re (cos a n ) > . 



Here the characteristic values sin 2 o; K and the 

 (normalized) characteristic functions U n (z) satisfy 

 the equation 



dHJ 

 dz 



J - + k n - [sin 2 a + M*\ U = , 



(3) 



a By Lt. W. F. Eberlein, TJSNR, Office of the Chief of Naval 

 Operations. 



plus the boundary and normalization conditions: 



(7(0) = (3a) 



eiutU(z) represents an outgoing wave for large 

 positive z . (3b) 



lim 



/ (*)—>- o 



U-dz = 1 



(3c) 



Usually sin 2 a is small, kd is large, and one has 

 Hf (kd cos a n ) 



2 



irkd COS a, 



dkw COS a, 



i(kd cos an) 



-ikd(l — (1/2) sin an) 



(4) 



The exponential decay factor of the horizontal waves 

 thus has the form 



exp 



-— I m (sin 2 



«n) > 



and the sin 2 « K values evidently lie in the upper half 

 of the complex plane. 



The problem is then to find the characteristic 

 values and characteristic functions of the system (3) 

 for a given dependence of modified index of refraction 

 upon height. For a ground-based duct of height h 

 with an M curve being made up of two line segments, 

 the upper having standard slope, equation (3) 

 becomes 



d 2 U 



dz 1 



+ fc 2 [A + y(z)} U = 



(3') 



where 



y(z) = 2ai(z - h) , < z < h 



y{z) = 2a 2 (z — h) , z>h , 



A = sin 2 a + 2aji , 



1 

 dz = — . 



The linear change of variable 



'k 



Xi 



a] 



[A + 2ai(z - h)] 



237 



