1162 
Maximum displacement. 
In the approximate formulae (23-25) the maximum displacement occurs when € = 0, i.e. when 
becomes very large. The maximum displacement is according to this forealey 2pémn? €, With 
inch plate exposed normally at 50 feet from 300 1b. of T.N.T., n= 2.3 x 10°, m= 5.0 gms/sq.cin., 
= 1 ton/sq.inch = 1.54 x 108 dynes/sq.cm., € = 12.1,this corresponds with a displacement of 
0.99 cm, It might therefore be supposed that if a structure of which a : inch plate forms an outer 
wall can be subjected to a sudden displacement of 0.99 cm, without injury it would necessarily be 
uninjured by the explosion of 300 1b. of T.N.T. at a distance of 50 feet, Such a deduction, though 
in agreement with the formulae so far developed, would probably not turn out to be justified in 
practice because the formulae assume that water can sustain tension as wel) as pressure. Referring 
to Figure 5 it will be seen that the positive pressure is maintained only for 1/10,000th sf a second. 
During this time the plate acquires a velocity of about 17 metres/second, and moves through a distance 
of about 0.13 cm. The slowing down of the plate is due chiefly to the suction phase which, though far 
less intense than the pressure phaso, continues for much longer. Ouring the slowing down process the 
i inch plate of Figure 5 suffers a further displacement of 0.958 - 0.13 = 0.83 cm. The maximum suction 
in this case is 0.12 x (maximum pressure in the incident pulse) i.e. 0.12 tons/square inch or 
269 1b./square inch or 18 atmospheres. It is this suction which in the foregoing theoretical treatment 
is responsible for the rapid deceleration of the plate. If the water is incapable of exerting suction 
the plate will leave the water as soon as the positive pressure vanishes. It is then moving at 
17 metres/second and the distance it will move before being brought to rest depends on the nature of 
the structure which supports it. If, for instance, the supporting structure is elastic and is of such 
stiffness that the plate would execute u/2.7 vibrations per second in the absence of the water (so that 
i has the meaning assigned to it in equation (6)) the equation connecting displacement and velocity is 
t 
1 
4 
Po 
é? fr Fd = a2 (28) 
where A is the amplitude of the vibration and is therefore the maximum displacement. If 'S5 is the 
displacement when tne plate leaves the water and So the velocity at this time 
at = ee a ate (29) 
If the frequency with which the G inch plate vibrates owing to its own stiffness and that of its 
supports is 100 cycles per second, “= 628. Using Ge = 0.13 cm, Se = 1700 cm/sec., A= (213)? + 
(2.7)? so that A= 2.7 cm. In this case therefore the maximum displacement is nearly three times 
as great as it would be if the water nad been assumed to be capable of exerting suction and 21 times 
as yreat as the displacement at the moment when it left the water. The displacement-time curve for 
the 1 inch plate when it leaves the water on attaining maximum velocity is shown in Figure 6. 
For this reason it is important to know what suction sea water will stand during the suction 
phase of a pulse and its reflection. Estimates based on the radius of the circle over which spray is 
thrown upwards when the pulse from a submarine explosion strikes the surface of the sea seem to show 
that water will stand a tension of about 200 or 300 1b./square incn for times of the order of 
1 millisecond. It does not seem to be certain, however, whether this tension could be mintained at 
the surface between water and, say, paint or iron. 
In connection with the formula (24) it seems worth while to give the expression for the 
displacement Se of a plate at the moment when the pressure changes to suction. In the case where the 
effect of the stiffness of the structure is neglected formula (24) gives 
2 3 € 
Be bees Dae + te, © 5 + (30) 
mn € 
In general this is small compared with the subsequent displacement whether the plate is assumed 
to leave the water or not. tn the latter case, when water is assumed to be capable of exerting suction 
it will be seen that the ratio 
E 
Displacement when pressure changes to suctj if ~“eé=1 
Maximum displacement whan arate Femains in contact with water ; 1-(e+ ie 
(31) 
Values of this ratio are given in column 8, Table 8, and are shown graphically in Figure 7. Its 
maximum value, 0.262, occurs when € = 1, 
Damage »eece 
