1175 
THE REFLECTION OF A SPHERICAL WAVE FROM AN 
INFINITE PLATE 
E. N. Fox 
1942 
* * * * * * ” * 
Introduction. 
Consider a Spherica) wave propagated from centre A in a medium of density P with wave— 
velocity C. The medium is bounded by an infinite plate POQ distance a from A, The wave is 
assumed to be of sufficiently small amplitude for the ordinary sound-wave equations to hold 
While the plate is assumed to move normal to itself and to offer purely inertia resistance to 
such motion. 
In the report "The pressure and impulse of submarine explosion waves on plates*, 
hereafter called Report A, G.1. Taylor considered a simllar plane wave problem with the more 
general boundary condition in which the plate offers also an elastic resistance to motion normal 
to itself, Sinee the frequency of vibration is usually relatively small in practice, Taylor 
gives approximate formulae which correspond to the exact solution for purely inertia resistance. 
The present problem is thus the spherical wave analogue of the plane wave problem whose solution 
is given by equations (22) to (27) of Report a. 
General Solution. 
\f A‘ be the image of A in the plane POQ then the problem is one of axial symmetry about 
AA’ and we shall use cylindrical co-ordinates x, r, with origin O and denote by R and R" the 
distancesof any point from A and A respectively. The zero of time t witl be taken to 
correspond with the front of the incident wave leaving A and we are concerned only with the 
region x < 0, 
Tne pressure p satisfies the usual wave equation 
Yh. sop leon 
oar Bee (1) 
while the incident wave from A is taken to be 
ane f (ct — R) (2) 
R 
Tne boundary equation at the plate is 
dv 
m— = ?p x 1=-8 10 
t (3) 
where m is the mss of the plate per unit area and v is its velocity assumed normal to itself. 
Nov in terms of the velocity-potential ¢ we have 
v= 2¢ 
Q x 
ad (4) 
bn: 
AND secves 
