Radar System Engineering

56 THE RADAR EQUATION [SEC. 2.14
and a diffusion of the vapor upward into the overlying air mass. This
implies the existence of a vertical gradient in the concentration of water
vapor with, normally, the highest concentration at the surface and
decreasing upward.
A typical condition to which such an effect may lead is that of a
relatively shallow layer just above the surface wit hin which the vertical
gradient of refractive index is negative and exceeds the critical value of
5 parts in 10’ per ft. Such a region is called a “duct” for reasons that
will appear shortly, and the level at which the gradient has just the critical
value is called the “top of the duct. ” Were we to trace the path of an
initially horizontal ray at this level, we would find it curving downward
just enough to keep up with the curvature of the earth, and therefore
maintaining constant height. Farther down in the duct, where the
gradient is stronger, it would be possible for a ray launched at a slight
upward inclination to be bent back to the surface again and thus to
proceed by a series of bounces, trapped as it were, within the duct.
From this temptingly graphic picture of propagation within a duct
it is easy to draw false conclusions. Since the description of the process
in terms of rays, traced by the rules of geometrical optics, nowhere
involves the wavekngth of the radiation, we should be led to expect
similar effects at all frequencies for which the index of refraction has the
same value, namely for all radio frequencies. But, actually, ducts such
as we have described have no observable eflect on the propagation of lowfrequency
radio waves. The reason for this is that the duct is effective
in “trapping” and guiding radiation only if the wavelength is less than
some critical value determined by the height of the duct and the steepness
of the gradient of refractive index within the duct. The ducts
which we have described have no great vertical extent, their heights
being of the order of a few tens, or at the most a few hundreds, of feet;
their influence on propagation is usually confined to frequencies in the
1000-M c/sec range and above.’ For gradients in refractive index which
would not be unusual in these surface ducts over water, the relation
between height of duct and the longest wavelength strongly affected by
the duct is suggested by the figures in Table 2.3.
TABLE2.3
Height totopof duct, ft . . . . . . . . . . . . . . . . 25 50 100 200 4@3
Longestwavelengthtrapped, cm*. . . . . . . 1.8 5 15 40 110
● Thesenumbersarebwedcmanarbitrary,althoughreavmable,criterionfortrapping,and“Pen a
simplifiedmodelin whichtherefractiveindexdecreasesupwardthroughtheductat thecomtantrate
of 8 partain 108perfoot. Theyareintendedonlyto beiUustrative.
1In certain parts of the world, the effect of trapping has been observed for frequenciesas
low as 200 Me/see.