370 
where A and B depend on the energy at the source. 
Neglecting the absorption along the path D (Fig. 5) 
the energy arrives on a zone of area Z perpendicular 
to the rays, 
Z = xlA? — (A — 6)*| cosi = 276A cos7. (31) 
The energy flux # at the distance A is then given by 
d cos 7 
C6 
pa Be? ( dA ie 
Z 6A cos 7 A 
di 
C(tan 7) — 
dA 
. @2) 
C is a constant depending on the energy at the source. 
The factor tan 7 is partly a consequence of the fact that 
near the source less energy passes through the zone a 
(Fig. 4) than through the larger zone 6 (see also Fig. 
6a). The greatest changes are produced by di/dA. If 2 
changes slowly with distance (Fig. 6c), the energy flux 
near B is relatively small. 
To simplify the picture, we assume that rays are 
parts of circles (7 = radius). Then cos 7 = A/2r, and 
equation (82) becomes in this special case [8] 
E* = (C/A®) [1 — (/r) (dr/da)|, (33) 
which shows the expected decrease in energy with the 
square of the distance. It is evident that in general the 
energy at a given point is especially large if the (mean) 
radius of curvature r decreases rapidly with distance A 
(or with the maximum height reached by the ray). 
According to equation (20), this requires that either 
the wind or the temperature or both increase rapidly 
with increasing height in the region of the highest point 
reached by the sound wave. Focal points (caustics) will 
result if di/dA becomes infinite; this would correspond 
to Fig. 6b, if two rays with small differences in 7 arrive 
at the same point B. On the other hand, the energy 
decreases considerably with distance if the highest 
parts of the rays enter a region where the radius of 
curvature increases rapidly, until the limit for straight 
rays is reached (equation (21)) and no energy arrives 
at the ground. For rays through the stratosphere, the 
absorption must be considered, and 
(tan 2) dt 
= SkdD 
EH = Ce K FING 
(34) 
INSTRUMENTS FOR THE RECORDING OF 
SOUND WAVES 
Most instruments used in recording sound waves 
through the atmosphere react to the change in pressure 
produced by the sound. In addition, records of distant 
explosions are sometimes obtained through seismo- 
graphs; in such instances the vibrations of the atmos- 
phere are transmitted by buildings or otherwise to the 
ground near the instrument. For sound-recording in- 
struments, membranes or pistons which form a part of 
an airtight contamer of a given volume of air are 
frequently used; their movements are magnified either 
by levers which operate a recording pen, by the mirror 
of an optical recording system, or through electro- 
THE UPPER ATMOSPHERE 
magnetic systems which record by means of galva- 
nometers. Even rather simple devices, such as a wire 
attached to the center of a wimdowpane and wound 
under tension around a thin needle carrying a mirror, 
have been used successfully for the recording of sound 
waves from distant explosions. The magnification of 
these types of instruments for long-period movements 
equals their. magnification for a constant continuous 
pressure; it decreases in the case of high damping to 
about 25 per cent for waves with half the free period of 
the vibrating system and exponentially toward zero 
for waves with still higher frequencies. If the damping 
ratio is less than 23:1, resonance increases the magni- 
fication near the free period of the instrument [9] and 
produces a maximum there of about twice the static 
magnification for a damping ratio of about 214:1. 
Many types of microbarographs with galvanometric 
recording have been developed in recent years. In 
Benioff’s microbarograph [13] a permanent-magnet 
moving-conductor type loud-speaker mounted in one 
of the sides of a sealed container is used as the re- 
sponding element. Output currents are recorded on 
standard seismograph galvanometric recorders. With a 
1.2 see galvanometer the maximum sensitivity is ap- 
proximately 1 mm deflection for 0.001 mb. In the 
microbarograph of Baird and Banwell [2] a diaphragm 
of “dulalium,” 0.005 mm thick, separates the body of 
the microphone into two compartments, one of which 
is closed to short-period pressure changes. Parallel to 
the diaphragm at a distance of about 0.015 mm is a 
brass disk which together with the diaphragm acts as 
a condenser whose capacity changes are magnified by 
electronic means and recorded by a galvanometer. The 
maximum sensitivity is about 1-mm deflection for less 
than 0.0001 mb. In the instruments designed and built 
at the Naval Ordnance Laboratory, Washington, D. C. 
[1] to record the subsonic waves from the Helgoland 
explosion in 1947, the flexing of a diaphragm in a 
microphone alters one of two matched inductive cir- 
cuits; the unbalance produces a signal voltage. This 
signal, the low-frequency modulation of an audiofre- 
quency carrier, is amplified and used to drive a modified 
Esterline-Angus graphic recorder for which a response 
curve has been given by Cox [4]. The maximum sensi- 
tivity is about 0.8-mm deflection for 0.001 mb. Other 
microbarographs, such as Macelwane’s, have their maxi- 
mum response for longer waves. 
Unfortunately, all these instruments respond not 
only to pressure waves, but also to pressure changes 
caused by air currents. If three instruments, forming a 
triangle with sides of somewhat less than one half the 
wave length of the sound waves, are used, the time 
differences between the instruments indicate whether 
the disturbance has travelled with the velocity of sound 
or with the much smaller velocity of air currents [3]. 
Simultaneously, the time differences between the pass- 
ing of a given point of the same wave at two pairs of 
stations makes it possible to calculate the direction 
from which the waves arrive and their angle of inci- 
dence [19]; the sound velocity must be calculated from 
the temperature. 
