A History of Research and a Review of Recent Developments

32
The detonation of explosive charges
The Mach stem is not always a straight line, but is usually assumed to be
so. As it travels outwards it acts in a similar way to the spherical shock front,
in that there is an instantaneous rise in overpressure. The Mach stem is initiated
when the angle of incidence (a) in Figure 2.6 exceeds 45°. In this figure we
have shown the reflected waves as semi-circular in cross section, but this is
not strictly correct because experiments show that the cross section more
nearly approaches the shape of a semi-ellipse.
The reflected shock wave travels faster than the incident shock wave because
the reflected overpressure is greater than the pressure existing in the incident
wave. This is a very fundamental aspect of the interaction of pressure waves
with surfaces which will be important later in the analysis of shock loads on
structures. The peak overpressure at the flat surface of the ground, p r, is related
to the peak incident overpressure, p 0, by the formula
p r /p 0 =2(7p a +4p 0 )/(7p a +p 0 ).
(2.2)
This relationship was first given in the form shown in Figure 2.7 for a value of
p a of 14.7 psi. It applies at zero incidence, i.e. when a=0 in Figure 2.6. The
formula does not change much as a is increased from 0 to 30°, but at higher
values there is a noticeable change, as shown in Figure 2.8. It is intriguing that
for low values of incident pressure (i.e. p 0=5 psi), the reflected pressure increases
as the angle of incidence changes from 40° to 55°. This is because blast waves
have finite amplitude, and the features of soundwave reflection (i.e. that incident
and reflected waves have equal strengths) no longer apply.
Figure 2.7 Relationship between peak reflected overpressure and peak incident
overpressure when p a =14.7 psi.