1160 
bree 
The expressions for p and & are 
2 S, (S, + 0) ' Ss, (s, + n) : Lae 
ae See DCL aE rte! mei es ec eit + ye ee aa eat. + (n? + us?) et 
Pp ee errr msin@ (s, - S,) msin@ (s, - 5.) 
2 msin@ 1 2 1 
(20) 
which is the same as (12) except that msin@ is substituted for m, and 
2p. sin@ 5. Saal 7 ees, ten 5 
ate Po La REAP) ieee! eSot (21) 
msin@ (n? - OE + yu”) S$, - 8, S,-S, 
which is similar to (13) except that msin @ is substituted for m and a factor sin@ has appeared, 
Numerical examples. 
The pulse from a submarine explosion of 300 1b, of T.N.T. falls to half value in 0.3 
milliseconds. This gives n= 2.3 x 10 sect, The motion which this pulse gives to a steel plate 
0.25 inches (= 0.635 cm.) thick will first be calculated. Here m= 5.0 grammes per sqecm, The 
velocity of sound in water is c = 1.4 x 10° cm, per second. Taking P= 1, pc/m= 2.8 x 10. In 
general yz is likely to be smal) compared with pc/m, even ifs for instance, the plate Is so rigidly 
supported that its period of vibration is 1/100th of a second, so that uz = 6.3 x 102, bm is only 
1/45th of pc/m. In these circunistances the approximate solution of (9) is Se pcim, S, = 
- umipc. This gives for the } inch plate 
4 bi 
Dae Mghegs (ore Olth =aianOlgier= 4 vu l0.te ena ono" 
Po 
and ‘ 
| 4 
Ge armas | sdazters- soos et) ene ex 107 
p ! 
ie) 
\f the same plate had been unsupported the motion would have been almost identical except 
that the terms 0.918 ee ¢ would be replaced by the constant 0.918. The displacement would have 
tended to a definite value of 0.667 x 10-8 Po (0.918). 
The pressure-time curve for points on the surface of the i inch plate is shown in Figure 5. 
Since p Is proportional to Py values of e/ Pp, depend only on t. The displacement of the plate is 
also shown in Figure 5, but in this case, though the displacement is also proportional to Por the 
result may be expressed more simply by assigning a definite value to po» The value chosen is 
1 ton/square inch or 1.5% x 10° dynes/sq.cm. This corresponds with the pressure found at 50 feet 
from a submarine explosion of 300 1b. of T.N.T. The displacement-time curve for i inch steel plate 
struck normally by this pressure wave is shown in Figure 5. It will be seen that the pressure 
vanishes and the maximum speed of 17 metres/second is attained after only 1/10th of a millisecond, 
At that time the displacement is only 1.3 mitlimetres, The pressure-time curve for the pulse is 
also shown in Figure 5, {n Figure 5 the time-scale is chosen so that only one millisecond is 
covered in order that the form of the pressure-time curve may be visible. The plate goes on moving 
for a considerable time after the attalnment of the maximum velocity. It will be seen later in fact 
that if the supporting structure exerted no restoring force tne plate woulg come to a stop at a 
definite limlting displacement. The restoring force makes it return slowly to its original position. 
The displacement-time curves are shown in Figure 6 for plates q inch and 1 inch thick, the restoring 
forces being such that in each case the plate would have a free period of 1/100th of a second when 
not in contact with the water. The : Inch plate reaches its maximum displacement of 0.96 cm. after 
two milliseconds, and after ten milliseconds it has only returned through 0.12 cm. to the value 
0.84 cm, Figure 6 also shows the time-displacement curve for a plate 1 inch thick. 
Approximate formula. 
If the frequency of free vibration of the plate is small compared witn the time constants n 
and pc/m, the formulae (20) and (21) can be written in approximate forms obtained by neglecting @ and 
writing S; = - (oc)/(msin@), Ss, = 0 , 
Poretee 
