304 
in keeping with both balloon and rocket measurements 
at White Sands, New Mexico, which likewise fail to 
show any genuinely isothermal layer [9]. 
Propagation of Sound Through the Upper Atmosphere 
Occasionally sound generated by an explosion can be 
heard distinctly at remote points, but not at all at 
points nearer to the source. The total region of audi- 
bility consists of a primary zone containing the source 
itself, plus one or more surrounding rings separated 
from each other and from the primary zone by regions 
of silence. Throughout the primary zone the sound is 
propagated at normal speeds. Within the secondary 
zones, however, the sound arrives much later than 
would be expected of waves propagated along the 
earth’s surface. 
Anomalous sound propagation was first noted in 
connection with gunfire at Queen Victoria’s funeral. 
It was soon concluded that waves reaching the annuli 
of abnormal audibility had traveled mto and through 
the upper atmosphere before returning to earth. An 
early explanation was that high-altitude winds caused 
the waves to bend back to the ground. The observed 
omnidirectionality of anomalous sound propagation, 
however, was not compatible with the wind theory. 
Von dem Borne proposed an explanation based on 
refraction of sound waves [3]. If atmospheric properties 
vary only with height, the refraction law is 
—— = VW 
COS e€ Mo (1) 
where for a given path V is constant. The quantities v 
and e¢ are respectively the local speed of sound and the 
angle between the ray path and the horizontal plane. 
The ray paths are straight, curve upward, or bend 
downward according as v is constant, decreasing, or 
increasing with altitude. Assuming refraction as the 
cause of anomalous propagation, one concludes that’ 
sound returning to earth from the upper atmosphere 
passed through a region in which its speed increased 
with altitude. Moreover, at the apex of its path, the 
wave front was moving horizontally at the speed V. 
The speed of sound through a gas is given by 
VS re? (2) 
where y is the ratio of the specific heat at constant 
pressure to that at constant volume, F is the universal 
gas constant, 17 the average molecular weight of the 
gas, and 7’ its absolute temperature. To explain anoma- 
lous sound propagation, von dem Borne postulated a 
decreasing value of 1 with height caused by increasing 
proportions of the lighter gases above the troposphere 
[3]. Data available at present, however, show far too 
small a change in composition to vary either M or + 
sufficiently to account for the observed refraction, which 
must accordingly be due to changes in the only remain- 
ing variable, namely temperature. 
In 1923 F. J. W. Whipple first suggested an explana- 
tion of anomalous sound propagation based on a posi- 
THE UPPER ATMOSPHERE 
tive temperature gradient in the upper atmosphere [19]. 
Since then Whipple, Gutenberg, Duckert, and numerous 
other authors have written extensively on the subject 
[5, 6, 8]. Whipple’s explanation is the one now generally 
accepted, and leads directly to a method for determin- 
ing upper-air temperatures. 
Referrg to Fig. 2, suppose that within a zone of 
abnormal audibility P, and P, are neighboring points 
in line with the sound source. At the moment the wave 
front reaches P;, it still has a distance RP» to travel 
before reaching Pp». If dt is the time interval between the 
arrival of the sound at P; and its arrival at Ps, and if 
UY is the speed of sound in the neighborhood of P, and 
Po», then RP» is equal to v% dt. Letting dS = P,P», one 
has: 
dS cos € = v dt; 
or, using equation (1): 
dS a Vo 
—- = |, 
dt COS 
(3) 
The quantity dS/dt is the apparent speed of sound 
along the earth between P, and P»2, and can be measured 
quite easily. Moreover, as shown by (3), the measured 
value of dS/dt is precisely the speed of sound V at the 
apex of its path through the upper atmosphere. The 
value of V can also be obtained by measuring the angle 
of arrival @ of the wave front. 
Once determined, V can be used in equation (2) to 
compute the temperature at the apex of the sound path. 
To associate the temperature so calculated with a 
specific height, the ray path must be traced out in order 
to determine the altitude of the apex. This can be done 
quite accurately at lower altitudes, using data from 
balloon observations. Assuming a positive temperature 
gradient, the remainder of the path in the upper atmos- 
phere is then constructed so as to fit the observed 
transit times for the sound and so as to join smoothly 
onto the lower segments. 
Temperatures determined by Gutenberg, and more 
recently by Cox, are shown in Figs. 3 and 4 [8, 5]. 
Between 40 and 60 km the temperatures quoted by 
Cox are considerably below those obtained by Guten- 
berg. Cox’s results are more in conformity with rocket 
findings at White Sands. A temperature maximum, such 
as that shown by Cox near 55 km, has always been 
observed in rocket flights. 
Winds may have a large effect upon temperatures 
deduced from sound observations. With this in mind 
