Radar System Engineering

SEC.2.14] L7UPERREFRACTION 57
That the height of the duct and the wavelength should be related,
and in such a manner that the heights involved are hundreds of times
greater than the corresponding wavelengths, may perhaps be made
plausible to the reader acquainted with propagation through waveguides.
The duct is, in a sense, a waveguide. Let us consider an oversimplified
model of a duct in which the index of refraction, nl, is constant from the
surface up to some height a, where it abruptly changes to nz (Fig. 2“15).
If nl exceeds n.zby some very small amount ~, both nl and nz being very
nearly unity, a wave incident on the boundary CD at a grazing angle a
wader than V% will experience total internal reflection. Under such
conditions the region between A B and CD can be regarded as the interior
of a waveguide bounded by two reflecting surfaces. But in an ordinary
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.
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., . . ,“
,.
,, . . . . . . .
. . ‘., ,..
.. ,.. ,. ,,. .
.,”. ‘.. .. ,’” ‘..
FIG.2.15.—Propagationwithina duct (overeimplified).
waveguide there is a basic relation between the width of the guide and the
angle of incidence of the plane waves into which the simpler waveguide
modes can be resolved. The longer the wavelength and the narrower
the guide, the larger the angle a according to a relation which for very
wide guides reduces to a = h/2u. But if, in our model sketched in Fig.
2.15, a becomes larger than @, total internal reflection no longer
occurs, and the energy leaks rapidly out of the guide. We might therefore
anticipate some such restriction as )i/2a < ~% or h2 < %azbfor an
effective duct. The form of this result is not inconsistent with the
figures quoted above which were based on a constant gradient of nl in
the duct, although it must be admitted that we have brutally oversimplified
a problem that abounds in mathematical difficulties and
subtleties.
Actually, no sharp distinction between trapping and standard propagation
can be drawn. Even our naive model suggests this; for we need