ATMOSPHERE AND THE ACOUSTIC EFFICIENCY OF FOG-SIGNAL MACHINERY. 221 
From equations (33) or (27) we are enabled to trace out the progress of a finite 
wave corresponding to given velocity conditions at the origin x — 0. As has already 
been mentioned, the wave will give rise to a “ discontinuity” after having travelled 
a distance which, in any given circumstances, may be determined from the complete 
solution which has just been given. Little is known either theoretically or experi¬ 
mentally regarding the state of affairs which exists in the wave in the neighbourhood 
of the discontinuity. It is not impossible, as suggested by Stokes, that the 
discontinuity may give rise to a species of reflected wave which will travel backwards 
towards the seat of the initial disturbance, thus complicating the state of affairs in 
the medium as represented by the original equations for finite waves. In applying 
these equations it is therefore necessary to assume that the waves, supposed to be 
propagated along a cylindrical tube, are completely absorbed by some mechanism 
before discontinuity sets in. It is interesting to note that Hadamard # has pointed 
out that, under conditions most likely to exist in practice, the discontinuity to which 
finite plane waves tend might give rise to the formation of vortices. It is easy to 
see from physical considerations that waves of large amplitude would tend to set up 
vortices in the neighbourhood of solid obstacles or of pre-existing eddies. Gaseous 
viscosity and thermal conductivity must play an important part in the sequence of 
events, but the inclusion of these factors in the equations of propagation complicates 
the problem beyond hope of solution. 
(iii.) Application of Preceding Theory to the Generation of Finite Waves by the 
Harmonic Motion of a Piston. 
It will be evident from the remarks already made that the conditions under which 
waves of large amplitude are generated and propagated have an important bearing 
on the theory and design of fog-signal sound generators. Owing to the limitations 
of the existing theory in leaving out of account thermal conduction and viscosity, it 
is desirable that the subject be studied from an experimental point of view. A 
simple apparatus capable of practical realization consists of a circular piston made to 
vibrate harmonically in a cylinder of sufficient length that the effect of the end open 
to the free atmosphere may be neglected. One of the first questions to be studied is 
the formation of the discontinuity. 
If the motion of the piston at x — 0 is given by 
u — u 0 sin 27 rnt, 
the velocity in the wave at distance x from the origin is given by 
xja 
u = u 0 sm '2-n 
t- 
Y±1 
i+i(y-lW ®}’" 1 
(34) 
* Hadamard, J., ‘Lemons sur la Propagation des Ondes,’ Paris, 1903. The same idea has also been 
advanced in another connection by Fessenden, R. A., ‘ Science,’ Oct. 17, 1913-. 
2 G 2 
