70 
Equation C5 relates the displacement of the plate to the pressure p, 
in the incident pulse, that is the pressure in open water in the absence 
of the target. It is convenient to use the exponential form™ for 
this pressure 
P; = PB & eee eee eee 200 eee eee (c6) 
Substituting in equation C5 
a’x dx aivia 
mm +Pco— +ker = t-] eco eee eve eee 
at” at = Ga 
Equation C7 can be solved for the initial conditions which correspond 
to the plate being at rest before the explosion, that is 
x = O 
ae when t = 0 sieie (@ Moleel ese Menme(CO)) 
For most practical cases to which the theory is all applicable, it is 
sufficiently accurate, in the first instance, to neglect the term kx 
in equation C7. The solution for the displacement at any time is then 
2Po -net — -nt 
x = pee € -i1+é6 -f£e eee coe (C9) 
mn (€-1)& 
where € is a non-dimensional quantity defined by 
c 
E =Fe ness Mamet tbe Setstech (CLO) 
Corresponding to the solution C9 for the displacement, the pressure on the 
plate can be obtained from equations C3, C6 and C9 in the form 
—nec -nt 
Ne hats oe = en -e \ An eee (C11) 
Now, whatever positive value be assigned to£, solution C11 indicates 
that the pressure acting on the plate will be initially 2p,., 
corresponding to instantaneous complete reflection and will then 
decrease to zero, followed by negative values which will ultimately 
tend to zero asymtotically. 
*The symbol p, is used rather than p,, of equation 10 to avoid 
confusion with the latter suffix and the symbol m for the mass per 
unit area of plating. 
