174 TECHNICAL SURVEY 
These expressions can be computed with the aid of 
the WPA Tables (unpublished) for the J functions 
with real argument. The curves in Figure 1 were 
computed down to values of 6 such that A» did not 
deviate appreciably from Aj. 
Conclusion. From the computed attenuation for a 
surface duct it appears that, for the first mode, 
when 5(= hk‘) is less than 1, trapping is less than 
2 per cent and that when 6 = 3 to 5 (depending on 
the negative gradient a), trapping is 98 per cent 
complete. There is therefore a ratner.narrow range 
ot values of the parameter 6 (1 to 4) within which a 
rapid transition takes place from a condition of 
negligible trapping to a condition of nearly complete 
trapping. This result may have a bearing on the 
observed fading which is associated with ducts. 
CALCULATIONS FOR THE SECOND 
AND HIGHER MODES 
OF THE BILINEAR MODEL 
Computationsof characteristic values and height - 
gain functions for the second and higher modes of 
a bilinear model M curve was carried on by the 
Analysis Section of Columbia University Wave Prop - 
ARENA 
ES i 
SUNS 
An-> 
uw 
a 
a SS 
SS Son 
i) ' 2 3 ES) 
a 
Fiaure 2. Characteristic values of D, for a bilinear 
model. s = —1. Dn = Bm +7 Am. 8} = ratio of slope 
of lower segment to standard slope = s°. g = height of 
joint in natural units. 
agation Group. The first mode of the bilinear model 
was treated at the Radiation Laboratory MIT.The 
computations were carried out with the aid of 
tables of h functions prepared by the Harvard 
Computation Laboratory,under the direction of 
Furry. The work discussed here is mainly on 
surface ducts in which the slope of the low seg- 
ment of the M curve is negative. Cases with pos- 
itive slopes of the low segment of the M curve 
have been tried but were found to involve func- 
tions which are beyond the range of existing tables. 
Some results on the characteristic values are shown 
in Figures 2 and 3 (for a definition of natural units 
see preceding articles). In Figure 2 the slope of the 
lower segment of the M curve is the negative of the 
standard slope, while in Figure 3 the ratio of the 
slope of the lower segment to the standard slope is 
—/8. The curves A; and B; for the first mode were 
computed at the Radiation Laboratory. An imagi- 
nary part A;, which is proportional to the horizontal 
attenuation (decrement), starts at g = 0 with a 
value appropriate for a standard atmosphere and 
decreases continuously as duct height g increases. 
Beyond g = 3 the first mode is completely trapped. 
The curve A» for the second mode decreases initially 
too but beyond g = 21s seen to level off to a constant 
limiting value. The real part of the characteristic 
value B, also approaches a constant limiting value 
for g greater than 3. These curves were obtained by 
solving the secular equation for D and also deter- 
mining the slope dD/dg at each point. The charac- 
teristic values curves can also be computed by 
starting first with Gamow’s values appropriate for 
large g and continuing backwards toward smaller 
values of g, being careful to determine the slope of 
the curves at each point. It is seen from Figures 2 
FIGuRE oh GR values Dm for a bilinear M 
curve. s = — V2.g = height of joint in natural units. 
s3 = ratio of slope of lower segment of M curve to the 
standard slope. Dm = Bm + % Am. 
