140 THE EEV. S. EAEXSHAW OX THE aiATHEMATICAL THEOET OF SOFXH. 
whose genesis is most violent. And we may also consider it as proved that those 
sounds will retain their original characteristics the longest which are the most gently 
generated. 
It is also quite evident from (13.) if the same cause generate sound-waves in different 
tubes fill ed with different gases, the wave will be soonest affected by the above modifi- 
cation in that tube which contains the gas for which (Jj is least. 
We come now to speak of 
2. The Wave of Marefaction. 
20. We shall obtain the equations for this kind of wave by wilting — U for -f-U in 
the equations of art. 12, which is equivalent to supposing a negative wave generated 
on the -\-y side of the piston. Hence the equations of a negative wave are 
^ 
x=\/ [/ jZ 
^ 
21. Seasoning in the same manner as in art. 14, it appears that the velocity with 
which the density p is transmitted is 
From this it appears that, speaking generally, the velocity of transmission of every part 
of a negative wave is less than of every part of a positive wave. The exceptions to this 
statement are the front and rear, which in both kinds of waves move with the same 
velocity s/ fjij, because for those points U=0. It is evident also that the most rarefied 
parts of a wave will be transmitted the most slowly, and will consequently di’op con- 
tinually towards the rear. Hence in this species of wave, as in the former, a constant 
change of type takes place ; and in the end also a negative or rarefied hore will be formed 
in the rear of the wave. 
By a process of reasoning analogous to that of art. 17, we infer that a negative 
sound-wave, from the moment that a tendency to discontinuity begins in its rear’, has 
the property of constantly shortening its rear, and by this means its rear travels faster 
than at the rate \/ gj ; and also as it progresses it is constantly casting off from its rear 
in a regressive direction a long continuous wave of a negative character. Aid. 18 also 
admits of easy modification to this kind of wave. 
22. The velocity of transmission of a negative wave being \/ gj — U, and the last term 
of this expression admitting of arbitrary increase, it is eHdent that '{J=\/gj is a critical 
value, and that the part of the wave corresponding to that value of U is stationary. 
The corresponding value of f is y- 
Every part of the wave where the density exceeds this travels forwards ; but the parts 
where the density is less than this are regressive ; hence a wave, as a whole, m which ^ 
