ATMOSPHERE AND THE ACOUSTIC EFFICIENCY OF FOG-SIGNAL MACHINERY. 225 
in Section 5 (ii). Whether such a form of apparatus would be more efficient than 
the best of existing types of sirens is a question which only a quantitative test can 
decide.* 
7. On the Theoretical Calculation of the Characteristics of Finite Waves 
EMITTED BY A COMPRESSED AlR SlREN. 
In the following section an attempt is made to establish the theory of a compressed 
air siren, combining the results derived for the adiabatic flow of air through an orifice 
with the formulas obtained in the preceding sections for the characteristics of aerial 
plane waves of finite amplitude. While such a theory must, in our present state of 
knowledge, be very imperfect owing to the neglect of many complex conditions met 
with in reality, the results thus obtained for what may be called an “ ideal siren ” 
may serve a useful purpose in giving in a general way an idea of the state of affairs 
which may be met with in practice. 
The essential features of a siren consist of a reservoir in which air is maintained at 
constant pressure p 1 and p 1} an orifice or series of orifices periodically opened and 
closed allowing the air to escape in intermittent puffs into the resonator, which we 
take for simplicity to be a long cylindrical tube of cross-section S. The pressure and 
density in the sound-waves a short distance from the ports of the siren we denote by 
( p , p) ; these are the pressures and densities in the finite wave generated in the 
cylindrical resonator, and are therefore connected with the velocity u in the wave by 
equation (26). We denote by (p 0 , p 0 ) atmospheric pressure and density. 
Applying Bernouilli’s equation to the steady adiabatic flow of a gas through an 
orifice,'we obtain for the velocity q at the low-pressure side of the orifice where the 
pressure is p 
This result assumes that the flow is stream-line, and that at a sufficient distance 
from the orifice oil the upstream side where the pressure is r Pi the velocity is 
negligible. Integrating (47) for adiabatic flow when p/pi = ( p/pi) y we obtain for the 
mass-flow the expression 
12 2 
Mp 
p\p\ 
7 
_y_ 
-i 
p_ 
pi 
(48) 
which is a well-known formula due to Saint-Venant and Wantzel. I 
If we denote by A (t) the total effective area of the ports and assume that formula 
* [Added February 14, 1919.—In the opinion of some practical fog-signal engineers the actual velocity 
of the air in the trumpet of a diaphone has an important effect in determining the loudness of .the signal 
and its atmospheric penetration.] 
t Lamb, ‘Hydrodynamics,’ 3rd edition, 1906, p. 23. The limitation of this formula forms the subject 
of a recent discussion by Rayleigh, ‘Phil. Mag.,’ vol. 32, Aug., 1916, pp. 177-187. 
