ATMOSPHERE AND THE ACOUSTIC EFFICIENCY OF FOG-SIGNAL MACHINERY. 217 
Inserting for |s| Boltzmann’s estimate, 6'5xl(U 8 , we obtain for the total flux 
of energy across a hemisphere of radius r = 18'4 km., 
[dW/dt] = i-x (3'32 x 10 4 ) 3 x(6'5x l(T 8 ) 2 x 0’0129 x 2 tt (l'84 x 10 6 ) 2 
= 222 x 10 7 ergs/sec. = 0'3 H.P. 
It will be noticed that the energy propagated as sound at this distance is a very 
small fraction of the power required to produce it; also that this estimate makes no 
assumption as to the mode of propagation. We are not justified, however, in taking 
this estimate of energy flux to hold for all distances as would be required by the 
inverse square law of propagation, and in particular for the proportion of power 
converted into sound at the generator itself: actual tests with the Webster phono¬ 
meter over long distances show that the inverse square law is not even approximately 
true in consequence of atmospheric refractions, while measurements of acoustic output 
by a special thermodynamical method indicate that relatively large amounts of power 
(about 2'4 H.P.) may be converted into sound at the vertex of the siren trumpet. 
The attenuation of energy-flux to quantities of the order of 0'3 H.P. at 18’4 km. is 
attributed by the writer to “ atmospheric ” losses, the greater part of which probably 
occur in the trumpet itself and in its immediate vicinity. The cause of these losses 
yet remains to be investigated, and will probably be found to be intimately associated 
with the question of the abnormal mode of propagation of waves of great intensity, 
a subject which we shall discuss briefly in the following sections. 
§ 5. Sound-waves of Finite Amplitude. 
(i.) Note on the Results of Previous Investigations. 
The equation for the propagation of sound-waves of small amplitude may be 
written in the familiar form 
d 2 y/dt 2 = a 2 d 2 y/dx 2 , .(17) 
where y represents the displacement of a layer of air from the equilibrium position. 
The derivation of this equation assumes that it is legitimate to neglect higher 
powers of the condensation s than the first. The exact equations governing the 
mode of propagation of waves of finite amplitude were first derived by Poisson* 
as long ago as 1808, and were discussed in further detail by Stokes! in 1848. In 
the case of adiabatic propagation, the exact equation was first derived by EarnshawJ 
(i860) in the form 
(dy/dx ) y+1 (d 2 y/dt 2 ) — d 2 d 2 y/dx 2 .( 18 ) 
* Poisson, “ Memoire sur la Theorie du Son,” ‘Journ. de l’Ecole Polytechnique,’ vol. VII., p. 319, 
et seq., 1808. 
t Stokes, “ On a Difficulty in the Theory of Sound,” ‘ Phil. Mag.,’ Nov., 1848 ‘ Mathematical and 
Physical Papers,’ vol. II., p. 51. 
1 Earnshaw, ‘Roy. Soc. Proc.,’ Jan. 6, 1859; ‘Phil. Trans. Roy. Soc.,’ 1860, p. 133. 
