THE EEY. S. EAENSHAW ON THE MATHEMATICAL THEOET OF SOUND. 137 
13. Now with respect to the genesis of this wave, we have seen that U must satisfy 
dy dY 
the same conditions as u, and Y as y. But therefore U=^: and again, as 
one of the equations of the general integral (7.) was obtained from the other by dif- 
ferentiation with regard to a, it follows that both a and U must vary continuously ; and 
that ^ must not pass through inhnity ; in other words, if the velocity of the piston 
vary it must vary continuously. Neither Y nor U must be discontinuous with regard 
to T. Hence there must be no discontinuity of pressure within the limits of the Avave 
at its genesis : and if discontinuity should afterwards occur in the wave during its trans- 
mission, our equations will cease to be applicable for that part of the wave where the 
discontinuity has occurred. For the wave in any one position may be supposed to 
generate its next position ; and a piston or diaphragm may at any time be supposed to 
act the part of the generating wave. What is necessary for the diaphragm to observe as 
a law of genesis must be necessary for the wave considered as the generator of its next 
position ; and therefore the part of the wave (if any) where discontinuity occurs will 
be beyond the reach of our equations. 
14. It has been shoAvn that the density which at the time t is at the distance y from 
the plane of reference, was generated at the time T when its distance from the same 
plane was Y. Hence it has been transmitted through the space y — Y in the time T, 
and consequently the A'elocity of its transmission (as appears from the first equation of 
(12.)) is a/^ + U. 
15. The waA’e as a whole is included between two points of it for each of which U = 0, 
and consequently for each of those points the velocity of transmission is y / Hence 
the wave as a whole is transmitted with this uniform velocity. But all the parts of the 
wave, with the exception of its front and rear, are transmitted with velocities greater 
than this, — with velocities dependent on their respective densities, tience every part 
of the wave, with the exception of its rear, is perpetually gaining on the front, and the 
result is a constant change of type , — the more condensed parts hurrying towards the 
front, with velocities greater as their densities are greater. This cannot go on perpe- 
tually without its happening at length that a hore (or tendency to a discontinuity of 
pressure) will be formed in front ; which will force its way, in violation of our equations, 
faster than at the rate oi s/ yj feet per second ; and consequently in experiments, made 
on sound at long distances from the origin of the sound-wave, we should expect the 
actual velocity obserr ed to be greater than \/ yj, especially if the sound be a violent one, 
generated with extreme force (see art. 17). 
16. We have seen that the velocity of transmission of the density f is \/ yo-^V. Now 
the velocity of the particles where the density is § is u, which we have shoAvn to be equal 
to U. In a certain sense we may consider the velocity u to be a wind-velocity in that 
part of the medium, and then rve have an indefinitely small disturbance at that point 
transmitted in that wind with the velocity \/ y, imposed upon the wind. In other 
