289 
PRESSURE-TIME CURVES FOR SUB-MARINE EXPLOSIONS 
(SECOND PAPER) 
W. G. Penney and H. K. Dasgupta 
London uly 1942 
* * * ne * * * * 
Summary. 
Numerical step-by-step calculations have been made on the development of an underwater 
disturbance caused by the spherical detonation of a T.W.T. charye of density 1.5. The theoretical 
pressure-time curves at a distance agree safisfactorily with those observed. If P tons/square inch 
1s the peak pressure at D feet froma charye W 1b., the theoretical values of P fit the formula 
/ 
ae exp { 0.274 w/3yp } 
The experimental values at a distance for charges of density 1.57 are about 10% higher. 
The discrepancy may be due to many causes, the two malin ones being (1) the initial densities were 
not quite the same, (2) the assumed pressure and velocity curves in the spherical detonation wave 
were not quite correct, and correspond with a chemical energy release of 800 calories per gramme 
(a better value would be 1000). 
The most Interesting result of the calculation is the energy distribution throughout the 
System. in the early stages, the water surrounding the charge picks up energy very rapioly; the 
outward velocity is about 1400 metres/second. Roughly 305 of the energy of the charge has been 
given irreversibly to the water as heat by the time that the peak pressure is 1 ton/square inch. 
This paper describes an attempt to improve on the calculations of the underwater pressure 
time curves described in the report "The Pressure-time Curve for Underwater Explosions® (hereafter 
called Report "A*"). The results, however, are very much the same as before. Agreement with 
experiment Is very good, having regard to the extreme difficulty of maintaining accuracy through 
a long serles of step-by-step calculations. 
The method used !s a modification of that used before, and for brevity we shall therefore 
write this paper on the understanding that it Is to be read in conjunction with Report "A", 
Certain Improvements in the values of the velocity of sound and the shock wave variables In water 
as functions of the pressure have been made. The initia? condittons in the gas bubble have now 
been taken to be those behind the spherical detonation wave in T.N,T., derived by Taylor In R.C-178. 
Considerable difficulty was experienced In jetting a start on the step-by-step calculations, because 
the pressure and velocity gradients at the detonation wave front are Infinite. After some cogitation 
we decided that the regime shown In Figure 2 must be Fouyhly right. The guiding principle which we 
used In averaging out the infinite gradients of the detonation wave front was to keep the energy 
balanced. Strictly speaking our Initial conditions should be considered to be those of Figure 2 
rather than those given by Taylor. 
Adiabatics and shock waves in water. 
Expressing pressures p In units 1000 kgm./sq.cm., and volumes v of & gm, of water inc.c., 
it was suggested In Report "A* that the adidbatics of water passing through the state T= 20°C at 
atmospheric pressure are given by the equation 
(o + 7.82) (v = 0.6807) = 2,493 (1) 
4 reconsideration of the data of Bridgeman (Proc.Amer.Acad.Sci,, Wish.66, 185, 1931) up to 
p= 12, and some further new data up to p = 50 which he has kindly supplied have led us to adopt a 
different equation, We now use 
v(p + 3.000) 38 = 4, 166 (8 
The cece 
