Chapter 19 



CHARACTERISTIC VALUES FOR THE FIRST MODE 

 FOR THE BILINEAR M CURVE a 



The model of an M curve composed of straight- 

 line segments suggested itself to workers at the 

 Radiation Laboratory early in 1944 as one in which 

 phase integral calculations could be carried out very 

 rapidly. At about the same time Lt. Comdr. Menzel 

 suggested the use of this model together with tables 

 of Hankel functions to obtain exact solutions. In the 

 fall of 1944 it became evident that phase integral 

 methods were not of much use with this model. 

 Tables of the required Hankel functions, essentially 

 standard height-gain functions, for complex argu- 

 ment were prepared at the Harvard Computation 

 Laboratory, and considerable effort was directed to 

 the obtaining of exact solutions. 



Work at the Radiation Laboratory has been largely 

 confined to the first mode for a curve composed of 

 two segments. This work has progressed largely 

 through the efforts of Miss Dodson and Miss Gill 

 and Howard and Parker. Dr. Pekeris of the Columbia 

 University Wave Propagation Group has been di- 

 recting work on the second mode. 



The units, notation, and model are given by the 

 following formulas and illustrated in Figure 1. 



Y»(4)**»2M«I0" S 



Figure 1. Models and units. 



H = (fc 2 g 2 )" 



AY 



2ir 



with 



H (feet) = 7.24 [X(cm)]i, 

 X 



L = 2{kq 2 )-i = 2 



2tt 



ff 



with L (thousand yds) = 6.69 [X(cm)] s , 



h _ d dHJ 

 H' X ~L'dz 



= u^--T>^ + ( Y + D )U = 0. 



The natural units of height and distance represent 

 two different compromises between wavelength X 

 and earth radius a, so that X -f- H -5- L -s- a form, 

 very roughly, a geometric progression. It is seen that 

 for microwaves, heights and distances occurring in 

 practice are fairly small numbers of natural units. 



The M curves are plotted in terms of the height z 

 in natural units and of a quantity Y which is simply 

 M multiplied by a suitable wavelength dependent 

 factor. The standard part of the curve then has 

 slope unity. In the bilinear model the anomaly 

 consists of a segment with slope s 3 times standard, 

 or, in these diagrams, simply slope s 3 . For negative s 

 there is a duct; s positive but less than 1 gives transi- 

 tional cases; and s greater than 1 gives substandard 

 cases. 



The essential quantity <fr used in calculating the 

 field is given by: 



2-\A _* 



^ = (e 



iiot — 2iri d/X— i7r/4 



L 



x~i X 



2/"^* + iB ™* UJzJ U m (z 2 ) 



a By W. II. Furry, Radiation Laboratory, MIT. 



The power density is equal to the free space power 

 density multiplied by ^ 2 d 2 . The characteristic values 

 are complex: D = B + iA. For the standard case: 

 Di = -1.17 + 2.02i. (For X = 10 cm this corre- 

 sponds to an attenuation of 1.22 db per thousand 

 yards.) ^ consists of three factors: one, that for a 

 plane wave, which can ordinarily be omitted; the 

 second, a constant factor which depends on wave- 

 length through L, the natural unit of distance [this 

 factor can be replaced by just 2\/V if x 2 ( = d 2 /L 2 ) 

 instead of d 2 is written in the first line] ; and finally 

 the critical factor written in terms of natural units 

 only and involving characteristic values and charac- 

 teristic functions. The imaginary parts of the charac- 

 teristic values are the coefficients of horizontal 

 attenuation, and the characteristic functions are the 

 height-gain functions. 



It is seen that for a typical microwave frequency 

 the horizontal attenuation of the first standard mode 

 (g = 0) is rather sizable. The plot of the height-gain 

 curve shows that if both transmitter and receiver 

 are at about 200 ft there is a gain of 50 to 60 db. 



228 



