222 RADIO WAVE PROPAGATION EXPERIMENTS 
to temperature deficit is equal to the ratio of humidity 
fluctuation to humidity deficit and very nearly equal 
to the ratio of M fluctuation to M deficit. 
GRAVITATIONAL WAVES AND 
TEMPERATURE INVERSIONS! 
It has been noted that the guided propagation of 
microwaves is often accompanied by deep fades with 
periods of the order of a few minutes. The sugges- 
tion has been made that these fluctuations may be 
associated with atmospheric wave motion which could 
make the top of the duct an undulating surface rather 
than a level one.?®?” Therefore, it seems desirable to 
discuss, from a meteorological point of view, the pos- 
sibility of the existence of such atmospheric waves 
and the physical characteristics of any which might 
exist. The purpose of this paper is to review and sum- 
marize the meteorological information which is avail- 
able concerning the subject. 
A theoretical consideration of the problem indicates 
that atmospheric wave motion can occur at any sur- 
face in the atmosphere where there is a rapid change 
in wind velocity with height and a stable stratification 
of temperature. Such conditions are best fulfilled at 
temperature inversions, which, it will be noted, usu- 
ally correspond to a rapid decrease with height of the 
index of refraction. The wind shear supplies the 
energy to set up the wave motion, in the same way in 
which waves are formed at the surface of the ocean. 
Gravitation acts as a stabilizing or restoring force. 
Hence, these waves are of a mixed shearing and 
gravitational type. The waves may be stable or un- 
stable, depending on their wavelength, on the density 
and wind speed differences between the two media, 
and on the lapse rates in the two media. For any given 
values of the density and wind velocity differences 
and of the lapse rates, there is a critical wavelength 
below which wave motion is unstable; that is, it dis- 
appears into turbulent eddies because of the shearing 
effect. All wavelengths above this critical value will 
remain stable because of the gravitational effect. 
Hence one may speak of the former as “shearing 
waves” and of the latter as “gravitational waves.” It 
is the stable or gravitational type with which we are 
concerned. 
These considerations hold for wavelengths up to 
about 500 km. For longer wavelengths the effect of 
the earth’s rotation must be considered. In this paper 
only the shorter wavelengths where this effect may 
be neglected will be discussed. 
A mathematical analysis of wave motion and deter- 
mination of the critical wavelengths involves a solu- 
tion of the equations of motion and continuity and an 
application of certain boundary conditions. In order 
iBy Lt. R. A. Craig, AAF, Weather Division. 
to derive the critical wavelengths given below, the 
following assumptions have been made. 
1. The inversion or shearing layer may be re- 
garded as a strict discontinuity between the air above 
and the air below. This assumption is sufficiently ac- 
curate provided the thickness of the layer is small 
compared to the wavelengths which occur. 
2. The velocities associated with the wave motion 
are small compared with the undisturbed velocities 
of the air masses above and below the inversion. 
3. The height of the inversion above the lower 
boundary (ground) is equal to or greater than 40 
per cent of the wavelength which occurs. 
4, There is no friction between the two fluids. 
Two cases may be considered. The first is the case 
where the air masses are assumed to be incompressible 
and homogeneous. It also holds for two air masses 
with adiabatic lapse rates. In this case the critical 
wavelength is given by*®* 
Qn (U-U)P TT 
(T'—T)g T+T 
In the second case the air masses are compressible 
and isothermal. For this case the critical wavelength 
is given by?® 
d crit = 
Vo ee OP 
g 4 
T +7 
\a- 1) +2 = 47) SO mi 
In these two eae 
T’ = temperature in the upper air, 
temperature in the lower air, 
velocity in the upper air, 
velocity in the lower air, 
acceleration of gravity, 
Cp/ Cy = 1.405, 
gas constant for air 
= 2.87 X 10° em?/sec? degree. 
In Table 13 the critical wavelengths in meters are 
tabulated for various values of wind shear and tem- 
perature difference. Values for the adiabatic case are 
tabulated above values for the isothermal case. For 
an intermediate lapse rate some intermediate value 
holds. 
Thus, for any given inversion, stable wave motion 
may exist so long as the wavelength is equal to or 
greater than the listed values and less than about 
500 km. There is no theoretical reason to believe that 
any particular wavelength in this wide range is more 
apt to occur in nature than any other. 
There is, however, some observational evidence to 
indicate that the wavelengths which occur in the at- 
mosphere are near the lowest possible values which 
can occur, namely, the critical values tabulated above. 
Billow clouds have been observed to occur near in- 
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