192 TECHNICAL SURVEY 
first mode by summing the series for he and perform- 
ing the integration numerically. This curve is remark- 
able for the considerable interval in which the 
ordinate is practically zero. The attenuation constant 
differs by less than 1 per cent from the standard for 
ducts below 6 = 1.2. Beyond this value the effect of 
the duct increases rapidly, and when 6 = 1.7 the 
attenuation constant is 10 per cent different from 
standard, and at 6 = 2 it is 20 per cent different. 
It seems that at least for radar purposes the 
condition 6 < 1 is a reasonable and convenient 
condition for defining a negligible duct. This is 
equivalent to saying that L/2 is the thickness below 
which a duct may for practical purposes be disre- 
garded. For instance, at \ = 10 cm, L = 33 ft, and 
hence we conclude that the effect of ducts less than 
16 ft in thickness on 10-cm radars may be neglected. 
On the other hand, if the wavelength is 3m, L = 300 
ft, and ducts below 150 ft in thickness are negligible. 
If in the interest of simplicity we neglect the effect 
of small variations in the characteristic values on 
the characteristic functions, the fractional change in 
attenuation constant is also equal to the fractional 
change in the range against surface targets. It follows 
that the estimation of range can be reduced to a 
measurement of sea temperature and specific hu- 
midity at masthead level; for the duct thickness d 
under conditions of neutral equilibrium is given by 
q; is the saturation specific humidity at sea tempera- 
ture and q, the specific humidity at masthead. I is 
a parameter for which a representative value is 
0.08, and (dq/dz)o is the gradient of specific humidity 
required to give zero M gradient under conditions 
of constant potential temperature. It is taken as 
14 g per kg. 
Thus it turns out that 
pr. Qs — Va 
6 = 0.32 aa 
is in grams per kg per 100 ft . 
h qs — Ve 
where L 
If L is given the appropriate value for \ = 10 cm 
6 = qs — Qa (g per kg) . 
For illustrative purposes, scales of (¢, — qa)/L and 
qs; — Ga for \ = 10 cm have been added in Figure 2. 
Bq/t  G/KG/100 FT 
1.0 
Aq G/KG 
Ficure 2. Fractional drop in attenuation constant of 
the first mode versus duct thickness. Bottom scale for 
Aqand A q/L corresponds to \ = 10 cm. 
It is emphasized that the calculations are rough 
and are presented only in the belief that some sort 
of simple guiding principle may be more useful than 
a highly accurate and cumbersome formula. The 
results given are accurate out to variations in range 
of 1 per cent, and the determination of threshold 
thickness is completely reliable. Extension beyond 
5 = 1.2 is a definite extrapolation. The trend indi- 
cating that the increase in range goes up at least as 
fast as the sixth power of the duct thickness for 
5 > 1 is, we believe, real. 
