222 DR. LOUIS YESSOT KING ON THE PROPAGATION OF SOUND IN THE FREE 
The discontinuity occurs for the least value of t for which dx/tiu = 0 : writing 
[l+|-(y—l) u/a], this condition may he written 3 x/dg = 0. 
In the application of these formulae to air, we take y = 1'40, so that (34) may 
be written 
x — at^—a^/(2-7rn) sin -1 [5 (l — g) ct/u 0 ']. . .... (35) 
The condition 3.r/3f = 0 leads to the equation 
x + ^a 2 g 7 /(2Trnu 0 )sec2,7rn(t — g~ 6 x/a) = 0,.(36) 
where f is given in terms of x and t by equation (35), from which it is seen that 
discontinuity will occur for some value of t for which 2-n (t — g^x/a) lies between -jtt 
and t-. In this interval it follows from equation (34) that u is positive, so that the 
minimum value of f is unity. Thus we may assert from (36) that the discontinuity 
will certainly not occur in the distance x given by 
x = %a 2 /(2 tt nu 0 ), . . .(37) 
this estimate agreeing with the lesser estimate of the two given in (23) when the 
numerical value for e = \ (y+ 1) = U20 is inserted therein. 
It is not without interest to calculate the rate at which the vibrating piston 
communicates energy to the atmosphere in the form of sound-waves. The rate at 
which the piston (of area S) does work is given by 
W = S pu = (p/p 0 ) y u, 
where u is the velocity of the piston and p is the density of the air over the section 
in contact with the piston. 
From the theory of finite ’waves the density is given in terms of the velocity in 
the wave by equation (26) in which we write U = 0, 
p/p* = .(38) 
so that 
w = p 0 S [l+i(y-l )u/a]y-i u, 
and the average rate at which work is done is given by the expression 
[W] =p 0 S y| 0 % {l+hiy-tfu/aj^dt. . ..... (39) 
Taking y = 1'40, the above expression may be evaluated in finite 
harmonic motion of the piston, u = u 0 sin (2 t mt). 
Expanding {l (y— l) u/a}y~ ] =( 1 -i- ^u/a) 7 , and integrating term 
obtain the expression 
[W] = y u S {p 0 u 0 2 /a) 
1 + UU^l\ 
% 
\5 a) 
\2 
+ 15 
' 8 
y 
5 a) 
+ 6 
>4 
«oY 
5 a 
terms for a 
by term, we 
(40) 
