ATMOSPHERE AND THE ACOUSTIC EFFICIENCY OF FOG-SIGNAL MACHINERY. 219 
or, in terms of the maximum condensation, jsj = XJ/d 
a / (Siren \ s |) < x x < a(l+e|s |)/(Siren j s |) .(24) 
By writing y = 1 in the above equations we obtain the values for the corresponding 
case of isothermal propagation given by Rayleigh.* 
(ii.) On a Contribution to the Theory of the Propagation of Aerial Plane Waves of 
Finite Amplitude. 
With a view to the future discussion of the theory of ideal sound-generators, we 
proceed to examine in greater detail the question of the propagation of aerial plane- 
waves of large amplitude. Following the main lines of Rayleigh’s exposition,! we 
denote by y and y + dyjdx . dx the actual positions at time t of neighbouring layers of 
air whose initial positions are defined by x and x + dx. The equation of continuity of 
mass gives us the relation p 0 = p dyjdx. If the expansions and condensations are 
supposed to take place according to the adiabatic law, p/p 0 — (p/po) y > we have 
p/po — (by/bx)~ y . The mass of unit area of the slice is p 0 dx, and the force acting on 
it is —(dp/dx) dx, so that the equation of motion, p 0 (d 2 y/dt 2 ) -!- (dp/dx) = 0, gives on 
eliminating p and writing a 2 = (p^y/pf 
(dy/dx) y+1 (d 2 y/dt 2 ) = a 2 d 2 y/dx\ 
which is Earn,shaw’s equation already quoted. Earnshaw’s solution of the exact 
equation proceeds on the assumption that there is at every point a definite relation 
between the particle velocity (u = dy/dt) and the density, p/p 0 = ( dyjdx ) _1 in the 
sound-wave, symbolized by the relation 
u — dy/dt — F (dy/dx). 
If we differentiate this equation with respect to t we obtain 
b 2 y/dt 2 = [r(dy/dx)J(d 2 y/dxf 
This equation may be identified with Earnshaw’s equation by choosing the 
arbitrary function F to satisfy the equation 
[F'(dy/dx)f = a 2 (dy/dx)-^\ 
or, writing for brevity a = (dyjdx) = pjp, F is determined from 
F / (a) = +aa~- {y+1) . 
* Rayleigh, ‘ Scientific Papers,’ vol. v., p. 575. 
t Rayleigh, ‘ Sound,’ 2nd edition, 1896, vol. II., pp. 31, et seq. 
VOL. CCXVIII.-A. 2 G 
