136 THE EEV. S. EAENSHAW OX THE 3IATHE3IATICAL THEOET OF SOEXH. 
this is, that a single distui’bance generates two waves; and (11.) shows that for one of 
them § is greater^ and for the other less than Pg. Equation (10.) shows that they are 
propagated in contrary directions on opposite sides of the piston, and are therefore not 
parts of the same wave. 
8. In the genesis of the wave we have supposed the piston pushed forwards, that is, 
in the direction of -\-x. Hence for the wave generated on that side of the piston we 
must, as appears from (10.), take the lower sign, which in (11.) gives p greater than p^. 
This wave we call \h.e jpositive wave, and the wave of condensation. For the wave gene- 
rated on the other side of the piston we must take the upper sign, which gives p less 
than ; and this wave we call the negative wave, and the wave of rarefaction. 
9. As it will be useful to have a definition of these two waves, which shall be uide- 
pendent of their position with regard to the generating piston, we may state that in 
general, — 
positive wave is one in which the motions of the particles are in the direction of 
wave-transmission : and 
a negative wave is one in which the motions of the particles are in a direction 
opposite to that of wave transmission. 
10. If g>, and g >2 be the densities of the air in contact with the piston before and 
behind at any moment, and if^, and be the corresponding pressures ; then from (11.) 
we have 
— _iL 
and 'm, 
and .-. f,^ 2 =fo; 
which may be thus expressed in words : — if a piston move in a tube, filled with air, in 
any manner whatever, the densities of the air in contact with it at its front and back 
are such that the equilibrium density is a mean proportional between them. And since 
p—gjP.^ we h.vsKe pp^—pl, which furnishes us with a similar property for the pressures on 
the piston. 
a u 
11. Since and^ 2 =^o 2 ~''^? it follows that the resistance to the motion of the 
piston (calling S its area) is 
{px—p^)S=^[i'''^—z~^i)po^. 
Hence in different gases, if Po be the same in all, those will offer the greatest resistance 
to the piston for which is the least. 
It will be convenient from this point to consider the two kinds of waves separately. 
1. The Wave of Condensation. 
12. The equations for this wave are 
J,=Y+(^/;;+u)(<-T) 
