Chapter 22 

 ATTENUATION DIAGRAMS FOR SURFACE DUCTS* 



The horizontal attenuation (decibels per unit 

 distance) of a signal under assumed propagation 

 conditions not only is of intrinsic interest but is one 

 of the simplest quantities to verify experimentally. 

 In terms of the wave equation formulation this 

 attenuation is proportional to the imaginary part of 

 the characteristic value associated with the mode 

 dominant in the region of space in question. No one 

 mode may necessarily be dominant, and in certain 

 regions a weakly attenuated mode may be outweighed 

 by a mode more strongly attenuated but with a 

 stronger initial excitation. Well inside the shadow 

 zone, however, the "first" or least attenuated mode 

 is frequently dominant. The results presented apply 

 primarily to this situation. 



The type of M curve considered is the bilinear 

 model in which the M curve consists of two straight- 

 line segments, the upper being assumed to possess 

 standard slope. This model M curve is completely 

 characterized by two parameters : duct thickness and 

 M deficit. Figures 2, 3 and 4 refer to three 

 frequencies (200, 3,000, and 10,000 mc) and super- 



-l ™q 



=0.036 FEET = —z- 



dh 2 



OUCT THICKNESS (h„) 



M-M -*- 



Figure 1. M-curve model. 



standard conditions corresponding to ranges of 1 to 

 100 M units in M deficit and 10 to 1,000 ft in duct 



thickness. 



The solid curves are contours of constant decibel 

 attenuation (of the first mode) per thousand yards. 

 One enters the diagram with given values of duct 

 thickness and M deficit and interpolates between these 

 contours to obtain the corresponding attenuation. 



The dashed curves are contours of constant "trap- 

 ping index" (number of classically trapped modes). 

 In terms of the standard notation their equation is 



M deficit 



10_ 6 



2 



qh a + 



9X 3 

 16ft 2 



a By Lt. William F. Eberlein, USNR, Office of the Chief 

 of Naval Operations. 



Their significance lies in that they furnish an indica- 

 tion of the number of modes other than the first that 

 must be taken into account. If, for example, the 

 bilinear M curve in question corresponds to a point 

 midway between the m = 1 and m = 2 contours, 

 the first mode is strongly trapped and the second 

 mode attenuation is reduced considerably below 

 standard. Which mode is dominant then depends 

 critically upon the heights of transmitter and 

 receiver, and the simple first mode picture becomes 

 incomplete except at great distances and small 

 heights. 



If one attempts to apply these results to simple 

 surface ducts differing from the idealized bilinear 

 model one should first approximate the actual M 

 curve by a bilinear curve and then enter the diagram 

 with the values of M deficit and duct thickness 

 corresponding to the idealized curve. How to make the 

 best bilinear approximation to a given M curve is 

 an important but still open question. 



Except for the 0.01- and 0.001-db contours, which 

 were computed by an asymptotic method, the attenu- 

 ation contours were cross-faired from preliminary 

 Radiation Laboratory calculations. The author 

 wishes to thank the Radiation Laboratory group, 

 and Doctors Freehafer and Furry in particular, for 

 permission to use their data. 



240 



