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National Advisory Committee for Aeronautics, Technical Notes - Free Oscillations of an Atmosphere in Which Temperature Increases Linearly with Height

It is shown that when the temperature in the atmosphere increases
linearly with height, the speed of propagation of long waves does not
approach a limit with increasing wave length, as in the case of an
atmosphere in which the temperature at great heights is assumed to be
constant or decreasing, but increases linearly with the period.

The group velocity ultimately also increases linearly with the period and
becomes equal to half the phase velocity. The region of maximum energy
of the oscillation is shifted to increasingly higher elevations as the
period is increased. Whereas, in an atmosphere where the temperature
gradient at great heights is negative or zero, the tides are similar to
those in a uniform ocean of equivalent depth H (about 8 km, depending
on the assumed vertical temperature distribution), the superposition of
an outer envelope with a positive temperature gradient introduces a
radical change into the nature of the tide. Insofar as it is still
legitimate to refer the tides in such an atmosphere to those in an ocean
of equivalent depth H, it can be said that H becomes a function of
the period, which increases indefinitely with the period. The bearing
of these results on the resonance theory of atmospheric tides is discussed.

The fact that the solar semidiurnal tide in the atmosphere is about
100 times larger than the equilibrium value led Kelvin to the conclusion
that the atmosphere possesses a free period of tidal oscillation of
12 solar hours. The resonance period of the atmosphere must be within
h minutes of half a solar day in order to account for the Observed ampli-
fication and for the fact that the moon, whose tidal force is more than
twice as large as that of the sun, excites a barometric oscillation having
an amplitude of only one—sixteenth that of the solar wave. If the vertical
temperature distribution in the atmosphere, assumed horizontally strati-
fied, is known, it is possible to compute its free period of tidal oscil—
lation following a method due to the work of G. I. Ehylor (reference 1).

According to this method one first determines the speed of propagation of
long waves .V in a flat atmosphere having the same vertical temperature
distribution.' The resonance period or periods of tidal oscillations of
the atmosphere are then identical with those of an ocean of depth H
enveloping the earth and are determined from the relation.