SOUND PROPAGATION 
if the wave length Z approaches the mean free path / 
of the molecules. Schrédinger [26] writes 
ip = MU/L (26) 
where 
F = 12 72x/3[(K — 1) AK-* + 4 BK-“]. 
K C,/¢y; A and B are constants depending on the 
heat conduction q of the gas, its coefficient of internal 
friction wu, the molecular velocity c, and the mean free 
path J of the molecules: 
Bel. (27) 
Schrédinger calculated from laboratory experiments F 
= 30.1; Kélzer [18] later used F = 33.0. Consequently, 
loss of energy # from absorption between two points 
close enough to each other so that the absorption can 
be assumed to be constant is given by 
In (#,/E,) = — 0.4343 Fl D/L? = — 0.0013 e”/8/L?. (28) 
The factor e”/8 is based on the assumption of constant 
temperature throughout the atmosphere, but the re- 
sulting mean free paths are within the limits given by 
the N.A.C.A. tables for the stratosphere (see Table I). 
q = Ac; u/c 
TasLe I. Mean FREE Pars (/ X 10-° em) or MOLECULES AT 
DIFFERENT LEVELS 
Height above sea level (i km) 
0 20 | 40 60 80 100 120 
(a) | 1.0 | 12 | 150} 1,800) 22,000} 270,000 | 3,300,000 
(b) | 0.6 | 15 | 580 | 14,000 | 200,000 |6,160,000 |66,000,000 
(ec) | 0.7 | 10 | 259 | 2,600 | 20,000 | 260,000 | 1,700,000 
(d) | 0.8] 8] 120 830 4,400 36,000 140,000 
(a) | = 10-5 eb’, 
(b) N.A.C.A. [21] tentative minimum temperatures. 
(ec) N.A.C.A. tentative standard temperatures. 
(d) N.A.C.A. tentative maximum temperatures (for h = 100 
and 120 km during the day). 
If it is assumed that the distance D is 1 km, that the 
free path of the molecules decreases exponentially with 
altitude h Gn km) from 10- cm at sea level, that the 
height of the homogeneous atmosphere is 8.0 km, and 
that the wave length L is measured in meters, 
In L = 0.27h — 1.443 — AUn|— In(Z,/E,)). (29) 
Equation (29) enables the calculation of LZ as a 
function of h for a given absorption. Results are plotted 
in Fig. 3, which can be used to find the height h above 
which waves of a given length L lose their energy 
by absorption too quickly to be observed. Figure 14 
shows that sound waves travel relatively long dis- 
tances near the top level of their path. If it is assumed 
that a loss of 10 per cent of the energy per kilometer 
over a distance of 100 km (which results in a reduc- 
tion of the energy to less than 10‘ and of the amplitude 
to less than 0.01) corresponds to the normal limit for 
the observation of such waves, it follows from Fig. 3 
that, even in the lower stratosphere, tones with fre- 
quencies above that of a soprano cannot travel very far 
without being absorbed almost completely. For waves 
with a frequency near that of the middle a (485 eps, 
IN THE ATMOSPHERE 369 
wave length at 0C about 34 m) the critical level is about 
25 km; for the lowest audible tones (16 cps, L near 
20 m) the critical level is below 80 km; and for waves 
with periods of 0.3 sec it is about 100 km. 
T (SEC AT 0°C)—> 
LOW TONES ii NOT AUDIBLE 
HIGH MIDDLE 
TONES A 
Sl ! 
L (METERS)—>— 
Fic. 3.—Fractions of sound energy which are lost by absorp- 
tion when the sound waves travel a distance of 1 km (assuming 
a temperature of 0C throughout the atmosphere) based on the 
research of Schrédinger [26]; h = height in km, ZL = wave 
length in meters, 7’ = period in seconds. 
The preceding equations give only the order of mag- 
nitude of the absorption. Fog, smoke, water droplets, 
etc., affect the absorption, and the equations do not 
hold for waves with very short wave lengths (less than 
one meter), for which the absorption increases faster 
than given by the equations, nor for waves with lengths 
over a few hundred meters, for which the wave length 
is an appreciable fraction of the height of the homo- 
geneous atmosphere. 
The amplitudes of recorded sound waves through 
the troposphere and the lower part of the stratosphere 
depend less on absorption than on the change in energy 
flux due to the change in size of the wave front. To 
find these effects [8] we suppose that there is no wind, 
Fic. 4. 
a 
L\ ne AN ERS () 
A B A B A B 
Fig. 6. 
and that sound waves produced at A in Fig. 4 start 
with the same energy in all directions. The energy 
flux Ey through a zone z between two cones formed by 
rays with angles of incidence 7 and 7 + e respectively is 
given (see Figs. 4 and 5) by 
Ey A [cos i—cos (@ + €)] = —B6 gicosd 
qa? 2) 
