ATMOSPHERE AND THE ACOUSTIC EFFICIENCY OF FOG-SIGNAL MACHINERY. 213 
medium may be considered as a perfectly elastic fluid for which the relation between 
pressure and volume (per unit mass) is expressed by the adiabatic law 
pv y = const, or p/p 0 = (p/ Po ) y .(l) 
where p 0 and p 0 refer to the pressure and density at standard temperature and 
pressure. It has long ago been verified by experiment that in the extremely rapid 
compressions and rarefactions which constitute sound-waves, equalization of the 
resulting inequalities of temperature by thermal conduction cannot take place with 
sufficient rapidity to bring about uniformity of temperature : the compressions and 
rarefactions may therefore be considered to take place under conditions of no heat- 
transfer, that is, under adiabatic conditions, y is a constant which for air has the 
value y = 1*414. It was first pointed out by Laplace that under these conditions 
the Newtonian formula for the velocity of sound, v/(£\,/p 0 )> should be modified to 
a = ^{ypo/po) .( 2 ) 
Applying the above formula to the propagation of sound in air at standard 
pressure and temperature, and inserting p 0 = 1*013 x 10 6 dynes/cm. 2 , p 0 = 1*293 x 10 -3 
gr./cm. 3 , we obtain for the calculated velocity of sound the value a = 332 metres/ 
sec. = 1089 feet/sec., in good agreement with observation. 
§ 2. Plane Waves or Small Amplitude. 
For convenience of reference we write down several formulae relating to the 
quantities employed to specify the state of motion in a plane sound-wave. Following 
Rayleigh’s* notation, we denote the velocity-potential at a time t and distance x in the 
direction of propagation of a harmonic train of waves by 
</> = A cos 2 tt ( x—at)/\ .(3) 
where A is a constant depending on the amplitude and A is the wave-length. If n be 
the frequency, we have the fundamental relation of wave-motion 
a — n\ .(4) 
If [cZW/ dt ] represent the average rate of propagation of energy across unit area of 
the wave-front it is shown that 
[. dW/dt ] = 2ir 2 A 2 p t) n 2 1 Oj, . 
while the maximum pressure variation or pressure amplitude | Sp | is given by 
| Sp | = 27rp 0 A n . 
* Rayleigh, ‘Theory of Sound,’ vol. II., p. 15 (1896). 
2 F 2 
(5) 
( 6 ) 
