214 DR. LOUIS YESSOT KING ON THE PROPAGATION OF SOUND IN THE FREE 
Hence expressing the rate of propagation of energy in terms of the pressure amplitude 
we have 
| Sp | 2 = 2ap 0 [dW/dt] .(7) 
It is convenient to specify the state of affairs in the medium at any instant by the 
condensation s defined by the equations 
s = Sp/p 0 = (l/y) Sp/p 0 . (8) 
The displacement amplitude in the medium [ £| is given by the formulae 
|£| = A/a = \Sp\/(2vnapo) = a\s\/(2-7rn), .(9) 
and the velocity displacement by 
|^| = 2ttA/x= \$p\/{ap 0 ) = .(lo) 
One of the most important results arising from the fact that waves of small 
amplitude are propagated with a constant velocity independent of their intensity is 
the application of the principle of superposition of vibrations. By the use of Fourier’s 
Theorem it is proved that a periodic disturbance of any wave form may be analysed 
into a number of simple harmonic waves whose sum gives rise to the complex 
disturbance considered. It is thus sufficient in the case of sound-waves of ordinary 
intensity to consider the propagation of a simple wave, as discussed in equations (3) 
to (10). The relative harmonic constituents of the complex wave preserve their 
relative amplitudes unaltered during propagation ; in other words, the quality of the 
sound is propagated to a distance without change. This fact is well illustrated by 
the everyday experience that the various notes from a number of musical instruments 
played simultaneously can be individually recognized over long distances. These 
points are emphasized because, as will be seen later, they no longer remain true in the 
case of waves of very great amplitude such as those emitted by a powerful fog siren. 
It may be mentioned in passing that sound-waves of ordinary intensity are 
propagated with extremely little dissipation of energy due to viscosity or heat- 
conduction, the calculations having been carried out by Stokes* as long ago as 1845 ; 
here again the circumstances are different in the case of very intense waves, the 
experiments to be discussed in the sequel indicating that sound-waves of sufficient 
power to be audible at great distances can only be generated at the expense of heavy 
atmospheric losses, especially in the immediate neighbourhood of the generator. 
§ 3. Spherical Waves of Small Amplitude. 
An important practical case of propagation of sound is that of waves emanating 
from a concentrated source. As in the case of plane waves we assume for simplicity 
* Stokes, ‘Cambridge Transactions,’ vol. VIII., p. 287, 1845. See Rayleigh’s ‘Sound,’ vol. II. 
p. 315 (1898). 
