27 111 
possible the weight of the piston and the extent of its movement. In the limit the 
piston becomes a flat plate held close up against the copper, so that the only 
movement is that which takes place during the actual crushing of the copper. The 
theory of this type of gauge requires separate consideration. 
Let A be the area of the plate exposed to the pressure in the water, r the 
resistance of the copper when the plate has been moved inwards a distance s, A the 
amount by which the copper is finally shortened, E the corresponding quantity of 
energy, as shown in the calibration table, and R the resistance of the copper when 
s = A. ‘The energy registered by the copper is equal to the work done by the 
pressure in moving the plate— 
E = | rds = A | pads. 
here are tivo cases to consider (compare Sarrau and Vieille, Comptes Rendus, 1882) ; 
suppose in the first place that the pressure rises so gradually to its maximum intensity 
P that the gauge is able to keep step with it, the resistance of the copper at each 
instant being equal to the pressure on the plate ; in this case obviously-— 
See et 
Pes = 
A 
The other extreme case is when the pressure rises instantaneously to its maximum 
intensity P and remains constant until the gauge has come to rest; in this case— 
[ pas AG 
so that— 
sdk Sasa 
Fae 
if P is expressed in tons per square inch, E in foot-pounds, A in square inches, and 
A in thousandths of an inch— 
ioe 
PES s+ 2 ho eee 
In the pressure wave from a submerged high-explosive charge the conditions 
approximate much more closely to the second of these two cases than to the first ; the 
pressure rises to its maximum intensity almost instantaneously, certainly in a time 
smaller than the time-constant of any gauge that it has been possible to construct; it 
does not however remain constant but rapidly commences to fall. In this case— 
[pds 
is approximately equal to P’A, P’ being the average pressure during the time of 
operation of the gauge; consequently equation (1) can be used, substituting P’ for P. 
The remaining part of the problem is to determine the time-constant T of the 
gauge, that is to say, the time during which the plate is in motion when subjected to 
a steady pressure. This depends on the inertia of the system, which is seated not 
only in the plate but also to some extent in the copper; it is easily shown that the 
mass of the plate must be added to one-third of the mass of the copper to get the 
total effective mass M. The equation of movement is— 
ds 2 
M 2 + r= pA. 
It is shown in Section 28 that for moderate crushings r = (approximately) r, + ks, 
where +, = 400 lbs. and k = 62 lbs. per 10° inch. Consequently— 
Assuming that the pressure is constant the solution is— 
k 
= io eee 
3s = 3A(1 cos »/* . t) 
The movement therefore ceases when— 
M 
k ’ 
t=T7 
