1043 
APPENDIX I 
THEORY OF PRESSURE IN FRONT OF AN AIR-BACKED FREE PLATE 
ACCELERATED BY & SHOCK WAVE 
When a plane shock wave strikes a free air-backed plate head on, the pressure in the 
water in front of the plate first rises as the original wave passes through it, then risen 
still further as the reflected shock wave returns from the plate. The pressure on the plate 
causes it to begin to move and its motion reduces the pressure, sending out a rarefaction 
wave. Consequently, the pressure out in front of the plate, after rising twice, then falls 
&nd actually goes negative, provided that cavitation does not interfere. The pressure can 
be calculated under certain simplifying assumptions; namely, the plate is assumed to be made 
of an infinitely rigid material, to be of infinite extent (but finite thickness), i.e., a 
rigid body whose motion is restricted only by its inertia. It is also assumed that cavita- 
tion does not form and that acoustic (small amplitude) theory can be used. 
The pressure at a point x (measured positively out into the water from the initial 
position of the plate) and at a time t (measured from the time the original wave strikes 
the plate) is the sum of three terms: 
(1) The pressure due to the original shock wave (assumed to be exponential 
in shape) 
t+% 
= (I-1) 
8 
Poe » fort poe , 
(2) The pressure due to the reflected wave 
coat 
t) 
Py @ » fort > $ ; (1-2) 
(3) The rarefaction wave from the motion of the plate 
- pou for t > = (I-3) 
In the above, p, is the peak pressure and @ the duration parameter of the 
original wave,~ gis the mass per unit area and y the instantaneous 
velocity of the plate, ¢ the velocity of sound in water, and the 
density of water. By Newton's law the sum of these pressures (acting at 
x = 0) will give the plate a velocity. 
us Bike aE, Fou (e“t/® -6 5 At/e) ’ (I-4) 
mn (4 -1) 
in which B =/ c0/n. Cea this value of u in the expression 
for the pressure at x the proper value of the time to use is t-x/ce because 
of the propagation time. It is convenient to measure pressure in terms of 
P,» time in terms of @, and distance in terms of c@; i.e., P = p/p,, T = t/0, 
= x/c@. Then the sum of the three terms becomes 
Pp =e“ (THX) . at en (T-X) + fa e-~ A(t-x) (1-5) 
an equation valid only for T 2X. These considerations navurally hold 
only so long as cavitation does not take place. 
Figure 131 shows some of the contour lines for P plotted against time T and distance 
61 15518 
