Chapter 24 

 SOME THEORETICAL RESULTS ON NONSTANDARD PROPAGATION" 



2,1 PROPAGATION IN THE OCEANIC 

 SURFACE DUCT 



The analysis section of Columbia University 

 Wave Propagation Group undertook a theoretical 

 study of propagation in case of surface ducts, which 

 have recently been reported to be of common oc- 

 currence in oceanic areas. The M curve chosen was 



M(h) = 346.4 + 0.036/j + 43e-°- 1 * , (1) 



where the height h is expressed in feet. This curve 

 has an M deficit of 43 units and a duct height of 48 

 ft and is considered to be representative of condi- 

 tions prevailing around Saipan when the wind is of 

 the order of 10 to 20 mph. 



The analysis was based on the phase integral 

 method. The standard W.K.B. (Wentzel-Kramers- 

 Brillouin) version of the asymptotic solutions of the 

 wave equation 107 had to be extended in two ways. 

 One was in the adoption of Langer's form of the 

 asymptotic solutions, 449 which enables one to bridge 

 the "gaps" around the turning points. The other, 

 and more important, development was in the exten- 

 sion of Langer's method to handle a case with two 

 turning points. This was accomplished by joining 

 the solutions from each turning point at the duct 

 height. The resulting solution agrees with Gamow's 

 for completely trapped modes but deviates from it 

 when leakage begins. For leaky modes the standard 

 Langer solution is adequate. 



Coverage diagrams were computed for the S and 

 X bands and for transmitter heights of 16 and 46 ft. 

 In case of the S band, it was found that the first 

 mode was nearly trapped, while the second mode 

 was considerably leaky with a decrement of about 

 3 db per nautical mile. The two modes were com- 

 bined, and coverage diagrams were computed over 

 ranges and heights such that the second mode 

 contributed no more than 25 per cent to the total 

 field. 



In the case of the X band, it was found that the 

 first two modes were completely trapped, the third 

 mode nearly trapped, while the fourth mode was 

 leaky with a decrement of over 3 db per nautical 

 mile. In computing the coverage diagrams for the X 



band, the four modes were combined over such 

 ranges and heights that the fourth mode did not 

 contribute more than 25 per cent to the total field. 



242 CHARACTERISTIC VALUES FOR A 

 CONTINUOUSLY VARYING MODIFIED 

 INDEX 



In the theoretical treatment of nonstandard prop- 

 agation by the method of normal modes, one is 

 confronted with the task of solving the differential 

 equation for the height-gain function U(h) given by 

 equation (2), which, it will be noted, is identical with 

 equation (8) in Chapter 25. 



t) m (h) + fc 2 [y(h) + AJ U m {h) = , 

 k = ~ , y{h) = 2 X 10~ 6 M{h) , 



A 



by asymptotic methods, the characteristic value 

 A m is determined, to a first approximation, by the 

 condition that 



k I \/y(h) - y{fn) dh = v m ^ir (m — -^J , (3) 



A m = -lAh) . (4) 



In order to solve equation (3) one has to find a 

 value hi, which is generally complex, such that when 

 y(hi) is substituted in the radicand and the integral 



1 



Vy(h) - y(hi) dh = F(hi) 



(5) 



a By C. L. Pekeris, Columbia University Wave Propagation 

 Group, Analysis Section. 



evaluated, the result should be purely real, and equal 

 to v m /k. In case of a surface duct, Fihi) is real and is 

 a continuously increasing function of its argument 

 for real values of hi ranging from zero up to the 

 duct height /i . In order for F(hi) to increase beyond 

 the value F(h ) and still to remain real it is found 

 that hi must be complex; i.e., the path in the complex 

 /ii-plane along which F(hi) is real consists of the 

 portion of real axis ^hi ^h Q followed by a curve 

 in the fourth quadrant. 



In case of substandard refraction, F(hi) is real 

 only for complex values of hi, and the method of 

 solving equation (3) to be explained presently is 



247 



