THEORY OF WAVES OF FINITE LONGITUDINAL DISTURBANCE. 
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pressures be equal to the undisturbed pressure ; that is, let 
P2 d~Pi p . 
2 x ’ 
( 20 ) 
then equations (18) and (19) become simply 
and 
« 2 =yPS ; 
( 21 ) 
( 22 ) 
the last of which is Laplace’s well-known law of the propagation of sound. The three 
equations of this section are applicable to an indefinitely long series of waves in which 
equal disturbances of pressure take place alternately in opposite directions. 
§11. Wave of Permanent Compression or Dilatation in a Tube of Perfect Gas . — To 
adapt equation (18) to the case of a wave of permanent compression or dilatation in a 
tube of perfect gas, the pressure at the front of the wave is to be made equal to the un- 
disturbed pressure, and the pressure at the back of the wave to the final or permanently 
altered pressure. Let the final pressure be denoted simply hypr, then^?,=P, and p 2 —p‘ 
giving for the square of the mass-velocity 
m 2 — s{(y+ 1 )f 2}’ 
(23) 
for the square of the linear velocity of advance 
a ‘ 
,nf S 2 =s{(y + l)f+( r -l)f}, 
and for the final velocity of disturbance 
S 
“=Uk P =(?-Vf 
[(7+ 1 )f + (5 /— 
(24) 
(25) 
Equations (23) and (24) show that a wave of condensation is propagated faster, and 
a wave of rarefaction slower, than a series of waves of oscillation. They further show 
that there is no upper limit to the velocity of propagation of a wave of condensation ; 
and also that to the velocity of propagation of a wave of rarefaction there is a lower 
limit, found by making y> = 0 in equations (23) and (24). The values of that lower 
limit, for the squares of the mass-velocity and linear velocity respectively, are as 
follows : — 
m''(/J=0)= (r ^A P ; (26) 
(27) 
and the corresponding value of the velocity of disturbance, being its negative limit, is 
“(i>=0) = -\/{^} (28) 
2 p 2 
