273 
THE PRESSURE-TIME CURVE FOR UNDERWATER EXPLOSIONS 
W. G. Penney 
November 1940 
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The purpose of this mathematical Investigation Is to estimate the tlme variation of the 
pressure at various distances from the centre of an explosion of a spherical charge of T.N.T. 
surrounded by water, The results of some measurements are avallable for comparison, and the 
agreement is satisfactory, Confidence may therefure be placed In the values glven by the theory 
for the pressures at positlons so close to the charge that the force of the explosion prevents 
measurements from being made. 
A rough description of the various events that occur is as follows. The explosive Is 
supposed to be detonated, and for want of more detailed and accurate information we assume that 
the resulting hot gases are instantaneously at rest, and that the temperature, pressure and 
chemical composition are uniform. Thus, at the Initial instant of time, we have a sphere of 
hot gas at a very high pressure, surrounded by water at atmospheric pressures A shock wave of 
great Intensity sets off into the water, and a rarefaction wave starts off into the gas. At 
the same time, the water and the gas in the immediate nelghbourhood of the gas-water Interface 
acquires a high velocity (of the order 1000 m/sec.). By the time the rarefaction wave has 
reached the centre of the gas, all parts of the gas are moving outwards, but the speeds In 
different spherical shells are different. Similarly, in the weter, varlous shells up to the 
shock wave front are moving et different speeds, and there is a considerable variation of 
pressure throughout the system. As the gas expands, it does work, and the temperature, and 
hence the chemical composition, change. The pressure everywhere drops, but nowhere so quickly 
as at the centre of the gas, the origin, immediately the rarefaction wave reaches there. The 
next stage, therefore, Is one where the pressure at the origin Is very low and rising more or 
less uniformly to a maximum at the shock wave front, The pressure gradient near the origin 
soon stops tne gas In this reglon from further expansion, and In fact soon causes yas to rush 
back again to the origin, restoring the pressure as it does so. The pressure gradient at the 
origin then becomes reversed, and gas rushes out again. fhe order of magnitude of the speeds 
at which the gas near the origin rushes in and out ts 200 or 30 m/sec. The effect of the 
pulsation of the gas In the central rcglon passes out Into the rest of the gas and thence into 
the water as rarefaction and compression waves. Another set >f waves is continually belng 
generated at the gas-water interface, and the net result of all these waves, going in and out, 
Is to average out most of them. Nevertheless, the pressure-time curve due to the explosion, 
even at considerable distances from the bomb, should show fluctuatlons. Because of the 
involved nature of the motion of the gas near the origin, it Is not possible without very 
lengthy investigation to proceed to times much In excess of 0.6 x 1077 seconds after the Initial 
instant. Fortunately, most of the events which interest us have occurred by then, and are 
being propagated outwards. 
One Interesting fact brought out by the calculations is the rapid decrease of the 
shock wave intensity with distance. If the taeory of sound were a valid approximation, the 
pressure drop at the shock wave front would vary like i/r, where r is the distance from the 
origin, and of course this willl hold for very large values of r. Figure 4 shows how the 
pressure drop at the shock wave depends on r, for values of r up to about 25 charge diameters. 
Our calculations may conveniently be separated into different sections. Section | 
describes a step-by-step method for dealing with a disturbance of spherical symmetry in a 
fluid. Section I! deals with the numerical construction of certain auxiliary functions 
formally Introduced In the previous section. Section 11! considers shock waves In water. 
Section IV glves the results of the calculations. Section V considers the extrapolation of 
the results to the position at which measurements were made, and compares the results with 
those observed. 
Sectlon lessees 
