368 
dW /dz = a, the condition for a curvature of the ray to- 
wards the ground is given by 
a [l + (W/C) sin? 7] > 
10yT-* [sin 7 — (W/C) cos? 7]. (21) 
Equation (21) shows that for our problem the lapse 
rate y and the change a of wind velocity with elevation 
are more important than the ratio of the wind velocity 
to the sound velocity (W/C). 
For very flat rays (¢ near 90°) equation (21) leads to 
a > 10yT-", (22) 
from which it follows that under average conditions 
in the lower troposphere (vy about 6C km, T = 290K) 
the wind must increase with elevation by more than 
about 344 m sec km in the direction of the sound 
propagation to bring the sound rays, which leave the 
source nearly horizontally, back to the surface. In the 
opposite direction the sound rays are curved upward 
more strongly than in still air (7 is smaller). 
The following are a few critical values of a for straight 
rays under the same assumptions for y and T and sup- 
posing that W/C is 0.1: 
Tf a exceeds these critical values, the ray curves down- 
ward in the wind direction; if a is smaller, the rays 
turn upward. It is noteworthy that in general the wave 
front is no longer perpendicular to the rays if sound 
is propagated in air in which the wind velocity changes 
with elevation. 
If the paths of the rays are close to a part of a circle, 
use of trigonometry shows that the maximum height 
H™* which is reached by a ray at the distance A is given 
approximately by 
H* = r{1 — [l — (/2r)]7} = A?/8r. (23) 
If, for example, a = 5 m sec? km, y = 6C km*+, 
T = 300K, Wo = 5m sec (which gives C = 348 m 
sec, r = 235 km), the following values result: 
10 km 100 km 
88.8° 77.6° 
54m 516 km 
The values in the last column would normally be use- 
less, since the assumption of circular rays is not ful- 
filled between the ground and an elevation of 5144 km. 
On the other hand, the results show that conditions 
favorable to good audibility of sound waves to dis- 
tances of many kilometers at the ground need not 
extend to great heights. 
. . ; 
The preceding equations can also be used to answer 
the question, How close must a sound source at an 
elevation h be in order that the sound can be heard at 
the ground? If, for example, the lapse rate is y and the 
wind is negligible, the maximum horizontal distance 
A* at which direct rays arrive at the ground is given 
approximately (assuming circular rays) by 
A* =a (Th/y)”, 
THE UPPER ATMOSPHERE 
which for y = 
14h”. 
All preceding results are based on the assumption 
that the change in pressure during the process is small 
compared with the pressure itself. For larger changes in 
pressure, the sound velocity is greater than given by 
the preceding equations. Since the theory is very com- 
plicated and has rarely been needed in problems of 
sound propagation in the atmosphere, it is not con- 
sidered here. 
Another assumption is that the wave length is small 
as compared with the height A of the homogeneous 
atmosphere, since otherwise dispersion results as an 
effect of gravity. Assuming that the gravitational con- 
stant g does not change with altitude (which does not 
hold for the higher parts of the stratosphere), Schrédin- 
ger [26] found that the velocity v of sound waves moving 
vertically upward is given approximately by 
vo/C = 1+ (1/32) (L/A)? = 1+ 0.003166 (L/A)?, (24) 
where L = wave length, C = sound velocity in quiet 
air and for waves moving horizontally; C is given by 
equation (7). For waves with periods of less than 1 
sec, the last term in equation (24) is smaller than 10°, 
and the effect of the dispersion (long waves travel 
faster) is negligible. In addition, i most practical 
cases the sound waves travel much closer to the hori- 
zontal than the vertical direction, which decreases the 
effect of dispersion. The group velocity for waves travel- 
ling upward is given by an equation similar to (24) 
except that the last term is negative. 
For long waves, equation (24) cannot be used. In 
addition to the effect of decrease in gravity with eleva- 
tion, free vibrations of the atmosphere affect the wave 
propagation. The theory for various models represent- 
ing the atmosphere has been developed by Pekeris 
[23]. In addition to the body waves discussed above, 
waves corresponding to free oscillations may be ex- 
cited within certain ranges of frequencies, depending on 
the assumed model. 
Energy of Sound Waves in the Atmosphere. The 
energy and amplitudes of sound waves in the atmos- 
phere with periods not exceeding a few seconds depend 
on two major effects: (1) absorption, and (2) changes 
due to the increase (or occasionally decrease) in the 
distance between rays (which controls the energy flux). 
The absorption of the energy is usually introduced 
by a factor e-*?, where k is the coefficient of absorption, 
supposed to be constant over the distance D along the 
ray. If k changes along the ray, kD has to be replaced 
by Jk dD. 
The propagation of sound is a molecular process. 
The velocity C of sound and the molecular velocity c 
are connected by the equation (see, for example, Schro- 
dinger [26]): 
6C km=1 and T = 289K gives A*¥ = 
(C/c? = K/3; (25) 
for air K = c,/c, = 1.403, and 
C = 0.6840c. 
As long as the molecules are relatively close together, 
absorption of sound remains small. It increases rapidly 
