THE EEY. S. EAEHSHAW ON THE MATHEMATICAL THEOEY OF SOUND. 143 
31. Suppose a portion of the tube to be filled with air of a different kind from that 
which fills the first part. Let §'o be the quantities for this air which correspond to 
of flio former ; and to prevent the two airs or gases from mixing, let them be 
supposed to be separated by an impenetrable film without weight and inertia. Then as 
there is equilibrium in the tube before the wave is generated, we have 
Let now a wave be generated in the first medium and transmitted towards the second ; 
then when it has reached the common boundary of the media, the velocities of the 
particles in contact with the film on both sides will always be equal. Let U' be this 
velocity at any moment, and U the velocity which the film would have had at that 
moment, if the second medium had been the same as the first. Then U — U' is the 
velocity lost by the particles of the first medium by the resistance due to their contact 
with the film. In other words, this velocity has been impressed on the particles of the 
first medium by the resistance of the film, in the reflex direction. This gives rise to a 
reflex wave in the fii’st medium, which we may consider superimposed on the wind of 
the original wave. And consequently if p be the pressm*e at the film due to the original 
wave, the pressure when this reflex wave has been superimposed, ^. e. the actual pressure 
U-U' 2U-U' U 
at the film, is , which 
But if we now turn to the other side of the film, the velocity U' has been impressed 
upon the particles of the second medium in contact with the film ; and hence the pressure 
of those particles on the film 
U' U' 
and consequently, as the pressures on the two sides are equal, we have 
U' 2U-U' 
a/ f// V/|X 
Hence the velocities of the particles at the film, for the incident, reflected and refracted 
waves, are respectively proportional to 
\/(^-\-\/ \/ and 2v^ f. 
There is nothing new in these formulae, except that they are here deduced without 
supposing the motions small. 
II. WAVE MOTION WHEN CHANGE OF TEMPEEATUEE IS NOT NEGLECTED. 
32. The heat developed by that change of temperature which is produced by the 
sudden alteration of density due to the passage of a wave, is probably taken account of 
by using the following equation as that which connects pressure and density. 
