188 
IV. EFFECTS OF PRESSURE WAVES 28 
8. ELASTICALLY SUPPORTED PLATE 
B= 1 (critically damped) 
22 mi te 
n#1: z= Tele i eter ite” 
re (yt) e “0 
B< 1 (underdamped) 
ai = aT In } ee PH! sin ("1 —B ut + tan? =F) 
Here tan Vi-B /(8—n) is to be taken in the first or second quadrant. 
The equations for x as written still contain the constant Hy» but this con- 
stant serves only to specify a time scale for the whole process. Otherwise all fea- 
tures are determined by the values of B andn. All of the equations represent the 
plate as returning ultimately to its position of equilibrium, as would be expected. 
A plot showing certain values of x,,/x,, the ratio of the maximum displace- 
ment 2,, of the plate to its static displacement x, under the initial pressure p,, is 
shown in Figure 13. Only a few points were calculated, because of the laboriousness 
of the work; these points are indicated 
by crosses on the plot. Based on these 
points, roughly correct contours were 
drawn by estimation corresponding to var- 
ious values of z,/xz,, as indicated near 
the contours. The abscissa of each point 
on the plot represents a value of manor 
orm for values above 1, however, dis- 
tances along the axis are laid off in 
proportion to the reciprocal of 7m, start- 
ing from O at the right-hand end of the 
plot. Similarly, the ordinate represents 
values of B up to 1, then of 1/8 from 1 
i to 0, i.e., of 8 from 1 toc. Values of 
x,/%, for points between the contours can 
Figure 13 be estimated by interpolation. 
