174 
MESSRS. A. E. H. LOVE AND E. B. PIDDUCK 
We have seen that the velocity of a junction relative to the medium is the value of 
II at the junction, and it follows that the value of r remains constant along a junction 
which travels in the direction of increase of x. If the junction travels in the opposite 
direction, the value of s at the junction remains constant. 
The motion consequent upon any initial conditions consists in the transfer of the 
existing values of r and s through the medium with the variable velocity already described. 
New values of r and s can be generated at boundaries and transferred through the 
medium. 
6. General Analysis of Compound Waves. —When the dependent and independent 
variables are interchanged in the equations 
^ + n|^ = o, 
dt dx 0 
dU T-r 0<T 
+n — 
dt 
dx„ 
= 0, 
there result the equations 
a t 
3X 0 TT 
du da 
= 0, 
da du 
- 0. 
The first of these shows that there exists a function Z of a and u which has the properties 
expressed by the equations 
x 0 = - II 
t = 
az 
du 
and then the second shows that Z satisfies the differential equation 
d_ 
da 
- 0. 
If Z can be found in accordance with this equation, the values of x t) and t answering 
to any simultaneous value of a and u can be deduced. 
There is a relation between Z and x, which can be obtained very simply by introducing 
for a moment a quantity m by the equation 
^ = a// 3 ’ 
for then we have 
-^=-^d P = dm, 
11 p 
and it follows that we have at once 
X,, — 
az 
CV3 
7T5 — 
dx 
dx t , 
t = 
u 
r X 
du 
dx 
dt ' 
and 
