SOUND PROPAGATION 
In a layer with a constant value of 
y = — dT/dz, (12) 
a) fee, (13) 
vy sin 2 
Since the lapse rate y is usually several degrees per 
kilometer, the radius of curvature of sound rays is in 
most instances of the order of 100 km. In the tropo- 
sphere, positive values of y (7.e., negative values of 
daT/dz) prevail, and the sound rays are turning up- 
ward; in layers of inversions and in the lower strato- 
sphere y is negative, and the sound rays are turning 
towards the earth’s surface in still air. 
The ray equation in Cartesian coordinates follows 
from Fig. 1 and equation (10): 
Be [(.%) 1)" 
M4 
Po _ 1] dz. 
ce flr Sin? 79 
Even in the simple case of a constant lapse rate the 
solution becomes rather complicated: 
a = (AY — Y?)* +24 sin {(2AY/A) — 1], (15) 
where 
(14) 
= aacllo 
= — Y=2-T7/y. 
y Sin? Ag’ 0/¥ 
For this reason, the radius of curvature is frequently 
used to find approximately the form of the ray, or tan 7 
is calculated from equation (10) and graphically 
integrated to find the horizontal distance x correspond- 
ing to a certain part of the ray (using the first part of 
equation (14)). The corresponding travel time ¢ follows 
from 
az 
ce Cecost 
dz 
t= 30 
C cos 2 
(16) 
The angle of incidence 7 at a given level, especially 
at the ground, can be found from Fig. 2, 
dw/dx C (dit/dx) C/E) 
where V is the “apparent velocity” of the sound ob- 
served in the horizontal direction (along x) and given 
sin 7 
Fic. 2—Sound waves arriving at the surface of the earth and 
angle of incidence 7. 
by the “‘travel-time curve” (time of arrival of a given 
impulse plotted against the distance <2). 
The apparent velocity V is equal to the true velocity 
C* at the highest point of the ray (if 7% = 90° and C, 
IN THE ATMOSPHERE 367 
= (*, in equation (10), we find sin 7 = C/C*; equation 
(17) then gives V = C*). The corresponding height 
z = H can be found from 
H= f | qdx where coshg = Vs 
T 
V, 
sin 7, 
sin 7, 
(18) 
This equation is based on a solution by Herglotz [17] 
which, in simplified form, can be found in any book 
on theoretical seismology. The quantities Va and 7a 
are the apparent velocity and angle of incidence re- 
spectively at a given level for the ray which arrives 
at the distance x A and has its highest pomt at 
z = H; V, and 7, are the corresponding values for all 
points between x = 0 and « = A. Hquation (18) can 
be used only if the travel-time curve is continuous in 
this interval. In this case it permits the calculation of 
H as a function of the distance x, while the velocities 
at the height H (and from them 7) are given by the 
value of V at the corresponding distances x. Since in 
most practical cases the temperature 7’ and the velocity 
C decrease with height in the troposphere, the method 
can not be applied there. In this case a reference height 
(the tropopause or a higher surface) must be selected 
above which the temperature increases with height, 
and the horizontal component for the path and the 
corresponding travel time ¢ for the parts of the ray 
below the reference height must be calculated from 
equations (14) and (16) and deducted from the total 
distance and travel time. Then equation (18) can be 
applied to the part of the ray which is above the 
reference height. 
Sound Rays in Moving Air. Equations for the paths 
of sound waves, in the case where the wind direction 
forms an angle with the horizontal component of the 
sound path, are too complicated to be useful [6] (even 
if it is assumed that the wind has no vertical com- 
ponent and that its direction does not change with 
height) since the sound path is no longer a plane curve. 
Usually the wind component in the plane of the sound 
wave is considered and the component perpendicular 
to it is neglected. 
If W = wind component in the direction of the 
sound wave, and C = sound velocity given by equation 
(7), then the ray equation is 
(C/sin 1) + W 
sin 2 
const, 
C/(Co/sin %) + Wo — W. (19) 
Neglecting squares and higher terms in W/C, the radius 
of curvature r of the ray is now given [8] by 
T= 
C+ 3W sinz 
10 W dT iy Nene 
VT (sin — > cos 21 + (1 + C sin) 
¢ dz 
where C and W are measured in meters per second and 
7 is given by equation (19). If —d7T/dz = v lapse 
rate, and the change of wind velocity with height, 
(20) 
