Chapter 26 



FIRST ORDER ESTIMATION OF RADAR RANGES 

 OVER THE OPEN OCEAN* 



The most striking nonstandard propagation 

 conditions are for the most part associated with 

 meteorological conditions which can exist only over 

 those portions of the sea which are contiguous to 

 extensive land masses. At large distances from the 

 coasts, however, low ducts exist which, though they 

 never produce strongly locked modes at the usual 

 radar frequencies, nevertheless modify radar ranges. 

 The problem of the low duct has the great advantage 

 that conditions are sufficiently near standard that 

 numei'ical solutions can be found in convenient form 

 by an extension of the perturbation methods of wave 

 mechanics. 



At appreciable distances from land the temperature 

 of the air is essentially that of the sea, and the air 

 is in neutral equilibrium. Montgomery has pointed 

 out that under these conditions there is much 

 evidence to support a logarithmic distribution of 

 specific humidity. 



The logarithmic distribution of water vapor leads 

 to an M curve given by 



M - M = 



10 6 



l) [l ' ln f] 



where d is the duct thickness, z is the height coordi- 

 nate, and a is the radius of the earth. If we plot the 

 function in the brackets, we obtain the dashed 

 curve of Figure 1. 



This type of M distribution is inconvenient because 

 (a) the logarithmic term which represents the modi- 

 fication does not approach zero as the height increases 

 as a modification term should; and (b) ln(z/d) 

 becomes infinite when 2 = 0. Accordingly it is pro- 

 posed to replace the function in the brackets by the 

 first two terms of its series expansion about the 

 minimum. This amounts to substituting for the 

 logarithmic curve a parabolic curve which has the 

 same minimum point and the same radius of curvature 

 at the minimum point as the original distribution. 

 At twice the duct height the parabola has a standard 

 slope, and it is continued from that point upward as 

 a straight line of this slope {AB in Figure 1). 



The modification term is now represented entirely 



a By J. E. Freehafer, Radiation Laboratory, MIT. 



by the departure of the parabola from the line 

 AB, i.e., 



M-M = flO^[l+i(|-l) 2 ], 0<|<2 



M - Mo = ?10 6 -^, 

 4 ad 



2 < ^ . 



When the duct is low, the modes leak and are not 

 far different from the standard ones. Thus it seems 



Figure 1. Schematic M curve for ground-based duct. 



reasonable to employ the well-known methods of 

 perturbation theory for calculating the characteristic 

 values and functions of the parabolic atmosphere in 

 terms of departures from standard. 



If we brush aside mathematical questions of a 

 delicate nature, it is possible to obtain an approxi- 

 mation for the characteristic values which leads to 

 the following expression for the fractional change in 

 the attenuation constant (i.e., the real part of y m ) 



Re (y m ) - Re (y m ) 

 Re (7m) 



- [ (f - 5) Im [V (f + e m )] rff 



■/" _ _5*_ 



d [In' (e m )] 2 Im (e m ) ' 315 



Here, Re and Im designate the real and imaginary 

 parts, and 



256 



