velocity relative to the air ahead of it. Here, we will consider the decay of the shock
wave numerically by using artificial viscosity. Before we do so, let us visualize this
catching-up process by referring to the x-t diagram shown in Fig. 4.49. In addition to
the shock and piston paths, a rarefaction wave is shown commencing at t ¼ t R when
the piston’s motion is arrested. The head of the rarefaction wave has slope dx/
dt ¼ u p + c, where c is the speed of sound in the medium behind the shock. This
speed is given by the equation, c ¼ c 0 + (γ À 1)(u p /2), where c 0 is the speed of sound
in the undisturbed medium. Referring to Fig. 4.49, the head of the rarefaction wave
intersects the shock path at the point (x 1 , t 1 ) and these coordinates can be obtained
from the following equations,
x 1 ¼ Ut 1
and
x 1 ¼ u p t R þ u p þ c
À
Á
t 1 À t R
ð
Þ:
Solving for x 1 and t 1 , we find that
t 1 ¼
ct R
u p þ c
À
Á À U
ð4:63Þ
and
x Ut
=
R
t
1
t
x
t
x = u p t
x R = u p t R
x 1
Piston
Motion
Rarefaction wave
Shock Path
Shock Wave
Fig. 4.49 Sketch in the x-t plane showing the rarefaction wave catching up with the shock wave
and diminishing its strength (see text)
190
4 Numerical Treatment of Plane Shocks