1516 
where the notation is that adopted by Lamb. This equation may 
be reduced to an ordinary differential equation in x by carrying 
out a Laplace transformation with respect to the time. The 
transformation is given by the reciprocal formulae” 
Plxsr) - | pl t) exp Crt) dt DRE pote ee, 
Oo pee 
pls t st Play) eplrdy , or (66) 
Ou bod 
The transforms of the derivatives are given by 
[% ep(-yt)dt : y Pest) - pl% 0) 
[ Beevers: P Plat) rite) geal © 
(o) 
The physical case of interest here is the propagation 
of a transient wave which vanishes everywhere for t £0 and is 
then a known function of the time at x = 0. These conditions 
require: 
plso) =° 
for x fo 
2 p(%)o) _ 
= aeaid 
(68) 
Then the propagation equation (64°) becomes, upon use of equations 
(67) and (68): 
Sx? (69) 
