185 
25 IV. EFFECTS OF PRESSURE WAVES 
7. ON A FREE PLATE 
wave bearing a rough resemblance to observed pressure waves (Figure 12). 
Exponential Wave 
Suppose thet 
p(t)= 0 for t<0 
p(t) = pier for t>0 
Then it is easy to verify by substitu- 
tion thet a solution of [10] is, for 
t>0, 
dz 2Do ( pe 
see SRG eo se-8:)) B= Ee 
dt pe—am Bi im Time 
This solution also satisfies the nec- Figure 12 
essary boundary condition that (dx/dt) 
= 0 at t = 0, the plate being at rest before the wave strikes it. (If pc =am, the 
solution is 
dz _ 2Po te-o' .) 
dt m 
Since the plate obviously comes to rest eventually, it follows from the con- 
servation of energy that the wave must be totally reflected from it. The total dis- 
placement of the plate is finite and equal to 
Eke 2p 
Az = == dt = S20) 
i dt apc 
This may be compared with the net displacement undergone by an unobstructed 
water particle as the incident wave passes over it, which is 
= Ly es (Pee Pec Te 
Juae peice rene dt = ope 
Thus we have the following important conclusions: 
1. The plate completely reflects the wave; 
2. The total displacement of the plate is finite and is just twice the dis- 
plecement produced in unobstructed water by the incident wave. 
These conclusions are independent of the mass of the plate. A heavy plate 
acquires a smaller velocity but retains it longer. It can be shown that the same 
conclusions hold for a wave of any form. Furthermore, it can be shown that the same 
conclusions should hold generally for any target provided that 
1. its characteristics vary only in one dimension, in the direction of inci- 
dence of the wave; 
2. there is a light medium, like air, beyond the target; 
