187 
27 Er EFFECTS OF PRESSURE WAVES 
8. ELASTICALLY SUPPORTED PLATE 
where in the present instance 
y= ae Hy = 27% [13a, b] 
Damped Free Oscillations 
If p =0, the plate can execute oscillations damped because of radiation 
of its energy into the water, the energy being carried away by compressive waves. 
The solution of [12] when p = O is, according to circumstances: 
if y>w, (overdamped):2 = A,e"!' + Aze”', y, = y+? — ub; ye = » — Vy? — HG 
2 
if ye; (underdamped ): a = Ae”'sin (2t%vt +a), v= VY Sa 
0 
Here A,, A,, A, @ denote arbitrary constants. 
Effect of a Pressure Wave 
If a pressure wave strikes such a plate, it is evident from conservation of 
energy that the final result must be complete reflection of the incident energy. The 
point of practical interest is the maximum displacement of the plate, to which cor- 
responds the maximum strain in the springs or other elastic support. The maximum 
displacement can be determined by solving Equation [12] if the pressure p in the in- 
cident wave is known as a function of the time. 
Consider, for example, the exponential type of wave already employed: 
p=0fort<0, p=pe"' for t>0 
Suppose that the plate is initially at rest and in equilibrium, with x = 0. The ap- 
propriate solution of [12] is most conveniently written in terms of the two auxil- 
iary constants 
=o tp 4 2 
x, = =f 2 = — 
8 mug fe Ho , Ho 
The constant x, represents the static displacement of the plate under the maximum 
pressure p, (as may be seen by putting in Equation [12] 2p= p, ,d’x/dt” = 0,dz/dt 
= 0). The value of 8 determines the character of the free oscillations; and n can 
be regarded as the ratio of the natural time scale of the plate to the time scale of 
the exponential wave. 
B>1 (overdamped) 
1— 2nptn?+ 0: 
ar ire). |S 
— Fp Or 
1-—2ngp+n* = 
aie Seal, ott ona e Prat [ln —Aret ge ey | 
