172 
MESSRS. A. E. H. LOVE AND F. B. PIDDUCK 
If s is constant, the second of these equations becomes an identity, and the first can 
be integrated in the form 
r = F (x 0 — Il£), 
where F denotes an arbitrary function. This can be proved easily, and the equation 
can be written 
X 0 -Ut = /(<r). 
In lilce manner, when r is constant, the first of the two equations becomes an identity, 
and the second can be integrated in the form 
Xo+Ut =f(a ■). 
A motion with constant r or constant s is described as a “ progressive wave.” A 
wave with constant s is propagated in the direction of increase of x, with velocity II, 
which depends upon the constant value of s and the local value of r. This is the velocity 
relative to the medium, not the velocity relative to the tube. Similar statements hold 
for a wave of constant r. 
4. Motion of a Junction .—When a wave is transmitted into gas at rest, or into a 
region where there is some other state of motion, there may be discontinuity in the 
values of the pressure, &c., in the two regions separated by the front of the wave. We 
consider here the case where there is no such discontinuity, but, while the pressure, 
&c., have the same values on the two sides of any plane x = const., the laws of variation 
of these quantities on the two sides of a wave-front are different. We describe such 
a moving wave-front as a “ junction.” Our immediate object is to determine the 
velocity of a junction relative to the medium. We shall attain this object by supposing 
that there are very slight differences between the values of any of the quantities on 
the two sides of the wave-front. 
Let w denote the velocity of the junction relative to the medium. In a very short 
time St a mass equal to p^w St has its motion and state changed from those specified 
by u, p, p, to those specified by u + A u, p -f- Ap, p -f A p. The increment of 
momentum must be equal to the impulse of the difference of pressure, and therefore 
we have the equation 
p 0 ww St A u = o) A p St. 
Further, the work done during the interval St by the external pressures on the ends 
of this element of mass must be equal to the sum of the increments of the kinetic and 
intrinsic energies of the element. Now the changes of state being adiabatic and very 
slight, the increment of the intrinsic energy per unit of mass may be put equal to 
-pA(l/ P ), 
and therefore we have the equation 
a>(p+Ap)(u + Au)St — apuSt = Tfp 0 wlV St {(m+Aw)‘-M 2 | —p 0 cowStpA(l/p). 
