A History of Research and a Review of Recent Developments

The decay of instantaneous overpressure in free air 19
input is equal to the gain in energy in the gas, so that, by conservation of
energy,
(1.17)
where v 1 and v 2 are the volumes occupied by unit mass, i.e. v 1=1/ρ 1, v 2=1/ρ 2.
Eq. (1.17) has become known as the Rankine-Hugoniot equation.
The above analysis leads to the general relationship between shock front
particle velocity, ū, and the speed of sound in air (u a), where u a=(8p a/ρ a) 1/2 and
ρ a is the density of the ambient air. This is
where p a is the pressure of the ambient air.
Also,
(1.18)
(1.19)
where ρ is the density of the air behind the shock front. Taking u a as 1117 ft/sec
and pa as 14.7 psi, Eq. (1.18) can be plotted as a relationship between u¯ and
peak overpressure (p 0) as shown in Figure 1.8.
The sudden discontinuous rise to p 0 is followed by a continuous decrease
until the pressure returns to atmospheric and p=0. The time between the arrival
of p 0 and the return to atmospheric pressure is the ‘positive duration’ (see
Figure 1.1), and for analytical purposes it is useful to represent the pressuretime
curve as a mathematical function. Two functions are often used:
p=p 0(1–t/t 0),
which is a simple triangular form, or more accurately,
(1.20)
(1.21)
where p is the pressure after any time t. By selecting a value for k (the
wave form parameter), various decay characteristics can be indicated.
Curves with very rapid decay characteristics are typical of nuclear
explosions, and curves with slower decay rates are typical of explosions
with large volumes of product gases. When k=1, the positive and negative
impulses (Figure 1.1) are equal, and the positive impulse is p 0t 0/e. The
curve given by Eq. (1.21) is often known as the Friedlander curve [1.19],
because it comes from the work of F.G. Friedlander on behalf of the UK
Home Office at the very beginning of the Second World War. Note that p/p 0
is non-dimensional and is therefore an intensity characteristic of the blast
wave system.