178 
MESSRS. A. E. H. LOVE AND F. R PIDDUCE 
The progressive wave generated at the piston M is determined by the equation of 
motion of this piston. This equation is 
M cu 
M a7 = - ( ’ y P> 
and it must hold at x 0 = 0 for all positive values of t. It may be written 
\ 2n + l 
r C U [ (X 
M—- = —«Po( — 
dr 
and, since in the progressive wave s is constant and equal to -br,,, or a- 
clt = — 
u = <r 0 , ]t 
gives 
M I rr \2»+l 
M / 5e) dc 
Mpo \er 
from which, since a = <x 0 when t — 0. we have at x 0 = 0 
t_ M<r 0 ;y„y p 
2nu>p 0 \\rr / J 
Put for brevity 
H = M cr 0 CT / 2 y , 
then we have the values of <r and / at x {] — 0 connected by the equation 
Now in the progressive wave we have 
."r 0 -IP =f(rr), 
where the function / is to be found from the condition that at x 0 = 0 the above relation 
holds between <r and t. Hence we find 
/w=- h { i -Q t 
and the progressive wave formula can lie written 
at + H fey- 
x 0 + R VJ 
In the motion described by these formulae any plane of particles, specified by a value 
of in the interval \c > x v > 0, remains at rest until t = x 0 jci, and then moves with a 
velocity u, which is equal to <x — er 0 . Therefore the value of x answering to these 
particles at any subsequent time is given by the equation 
