A History of Research and a Review of Recent Developments

18
1.4 THE DECAY OF INSTANTANEOUS OVERPRESSURE IN FREE AIR
When Rankine, and later Hugoniot, analysed the pressure, velocity and density
of a gas after the passage of a shock wave, they considered the conservation
of mass, energy and momentum before and after the passage of the instantaneous
jump in pressure. Their analyses are compared in ref. 2.38.
Suppose that in front of a shock wave travelling with velocity ū the pressure,
density and material velocity of the gas is p1, ρ1 and u1, and that after its
passage these quantities change to p2, ρ2 and u2. Then, by conservation of
mass,
and by conservation of momentum
Eliminating u 2 gives
The nature of explosions
ρ 2 (ū–u 2 )=ρ 2 (ū–u 1 ),
p 1 –p 2 =ρ 2 (ū–u 2 )(u 1 –u2).
(1.14)
(1.15)
(1.16)
which indicates that when, as is usual, the air in front of the shock wave is at
rest, u 1=0 and the velocity of propagation ū can be determined entirely in
terms of the pressures and densities on either side of the discontinuity. As the
compression of the gas in the shock front is very fast, it is reasonable to assume
that compression follows the adiabatic law (changes in pressure and volume
with no change in absolute temperature). Then we may assume that the energy
Figure 1.7 The prediction of Erode for a bare sphere of TNT of loading density 1.5
g/cm 3 (from Brode, ref. 1.18).