SOUND PROPAGATION IN THE ATMOSPHERE 
By B. GUTENBERG 
California Institute of Technology 
THEORY OF SOUND WAVES IN GASES 
Velocity of Sound. The velocity C with which a 
change in volume is propagated in a material with 
Lame’s constants ) and w and with the density p is 
given by 
C? = (’ + 2u)/p. (1) 
In gases, the rigidity » is practically zero, and the 
constant \ is equal to the bulk modulus k. Therefore 
equation (1) reduces to 
C? = bia k = —(dp/dv)v, (2) 
and p = pressure, » = volume. Newton assumed that 
the propagation of sound is an isothermal process, but 
disagreement between observations and calculations 
under this assumption led Laplace to suppose that it is 
an adiabatic process. Except for instances where the 
change in pressure is large as compared with the pres- 
sure itself, results calculated under this assumption agree 
with the observations. 
If » = volume and p = pressure, K = c,/c,, where 
Cp = specific heat at constant pressure, c, = specific 
heat at constant volume, the condition for an adiabatic 
process is 
where 
pv= = const or dp/dv = —Kp/v. (3) 
Introducing this in equation (2), we obtain 
k = Kp and 2 = Kp/p. (4) 
It has been assumed in equation (3) that K does not 
depend on the pressure. In general, this approximation 
is sufficient; for a better approximation, see Hardy 
and others [15]. If 7 is the absolute temperature and R 
the gas constant of the specific gas, 
R = poa/po = p/peT, (5) 
where a = 1/273.18 and p = density of the gas at 
OC and a pressure of 1 atmosphere; if Co is the corre- 
sponding sound velocity, equation (4) can be written 
C? = KRT = CeaT Ce = KR/a. (6) 
For dry air K = 1.4083, R = 2.87 X 10° em? sec 
deg™ which gives Cy) = 331.6 m sec—!. From a critical 
discussion of observations, Hardy and others [15] found 
331.46 + 0.05 m sec. With this value equation (6) 
becomes 
where 
C = 20.06+/T (m sec"). (7) 
Experiments did not disclose any appreciable differ- 
ence in the sound velocity for frequencies between 
10 and over 10° cycles. However, at very low and 
very high temperatures the observed velocity is smaller 
than the value calculated from equation (7). At —100C 
the difference is about 2 m sec, at +1000C about 
10 m sec7?. Quigley [25] suggested the following em- 
pirical equation from observations at low tempera- 
tures: 
@? = 387.62T + 1804307 ‘ 
— 2036400072 + 806 + 0.0300772, ©) 
which gives C = 330.6 m sec“! for 7 = 273.1K. 
In humid air the sound velocity is higher than in 
dry air by an amount H given approximately [8] by 
H = 0.14Ch/p, (9) 
where h = partial pressure (or tension) of the water 
vapor. For warm, humid air near the ground H may 
be as high as 5 m sect. 
Sound Rays in Quiet Air. The ray equation for sound 
waves in still air follows from Snell’s law and equa- 
tion (7): 
sint — Ci _ eh 
sin As Cy T. : 
in which 7 = angle of incidence (between the ray and 
a vertical line) at the points 1 and 2, where the wave 
velocities are C; and Cy; C and T are supposed to be a 
function of height z only. 
(10) 
Fre. 1.—Path of sound waves, angles of ineidence z, and radius 
of curvature r. 
The radius of curvature r of the ray can be found 
from Fig. 1 and equation (10): 
ds/di = dz/(cos 7) di = dz/d(sin 2), 
“al d(sin 2) i snidC _ dT smi 
dz CGH Ge Bk” 
r 
(11) 
Sle 
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