ATMOSPHERE AND THE ACOUSTIC EFFICIENCY OF FOG-SIGNAL MACHINERY. 223 
Even when u 0 approaches half the velocity of sound, this expression differs by only 
some four per cent, from the corresponding expression for waves of small amplitude 
as given by equations (7) and (lO), to which (40) reduces when uja is negligible. 
§ 6. On the Thermodynamic Basis for the Measurement of the Acoustic 
Output of a Compressed Air Siren. 
It is manifest from what has been said in the preceding sections that the theory of 
finite waves gives us no basis according to which the acoustic output of a siren may 
be calculated. We may, however, develop experimental means of estimating the 
rate at which energy is converted into sound. If we denote by M the total rate 
of air consumption, we suppose that a certain part m is utilized in the production 
of external work propagated away as sound, and that the work so done is 
performed adiabatically from pressure, density and absolute temperature p x , p x and 
0 } , respectively, to atmospheric conditions represented by p 0 , p 0 and 0 O . The calcula¬ 
tion of acoustic output will depend on the particular mechanism by which the 
utilizable air is allowed to perform external work. That representing conditions in 
a siren may be represented as follows:—AVe suppose the ports to be suddenly 
opened, allowing a volume i\ of air under conditions (p x , p u 0j) to pass into the 
resonator. AVe then suppose the ports to close while the volume of air v x expands 
adiabatically to a volume v 0 under atmospheric conditions (p 0 , p 0 , 0 O ), performing 
external work in compressing the layers of air in the resonator ahead of it, thus 
generating a single sound-wave. This cycle of operations is supposed to be repeated 
n times a second, so that in terms of the effective mass-flow 
77 1 = n Pl V x = Tlp 0 V 0 .(41) 
The remainder of the air-consumption (M— m) may be taken to include continuous 
leakage of air between the siren-cylinders, and that part of the intermittent flow 
through the ports which is in a violent state of eddy-motion. This eddy-motion 
probably dies out an appreciable fraction of a period after the ports are closed, so that 
the full expansion resulting from this portion of the flow is developed out of phase 
with the main wave and contributes nothing on the average to the energy in the 
wave. This portion will be referred to as “ leakage.’ The work done per cycle by 
the volume v x of air is 
Since the work is performed adiabatically, the temperature of this volume of air after 
expansion is given by (ni/n 0 ) 7_1 — 0 o /0!, so that the rate at which work is done in the 
n cycles per second may be written in either of the forms 
tv — 
(P°)*T 
y-lL \pj 
Pi 
pi y-l 
( 0 ,- 00 ), . 
making use of (41) and the adiabatic relation p x v x = p 0 v 0 y . 
(42) 
