180 
MESSRS. A. E. H. LOVE AND F. B. PIDDUCK 
x ----- x 0 + — (c + h—x 0 ) 
a 
2 n 
L l \ o- / 
2n—1 IW 
O”0 
2n—1 
which holds so long as the plane of particles is in the region occupied by the progressive 
wave. In particular, the displacement of the piston m is given by the equation 
2 n / \ h ( 2n \ 
■ = C + - - -((r u -(r)- - --- <r-<r 0 t, 
2 / 1—1 a \2n—\ 
in which 
rr ! 
The formula for Z is found to be 
(2n— l) a 0 lAo-/ 
a 
The First Middle Wave. 
12. Conditions satisfied at the Receding Front .—In the progressive wave from the 
left s is constant and r variable. The greatest value of r, which is the undisturbed value 
1<t 0 , travels at the front of the wave, and continually diminishing values of r, generated 
at the piston M, travel after it. Similar statements, with r and 5 interchanged, hold 
for the progressive wave from the right. The fronts of both waves travel along the 
tube with velocity a. When they reach the middle section, a compound wave begins 
to be generated there, and transmitted in both directions, encroaching upon the original 
progressive waves. This wave has a receding front, along which s is constant, travelling 
towards the left, and an advancing front, along which r is constant, travelling towards 
the right. The constant values of r and s at the two fronts are equal, and each of them 
IS -Tyc y. 
At the receding front the variations of x 0 and t are connected by the equation 
clx t] + II dt = 0, 
while the values of x u , t and a are connected by the progressive wave formula, which 
can be written 
.r„-ID + H (l — 
so that the variations of x 0 , t and II are connected by the equation 
day— (II dt-\-t dll) 
— cZII = 0. 
a 
