waves are calculated independently of the shock waves and the time of intersection
of the simple wave with the shock path is then determined. We will follow
Friedrichs’ analysis as presented below.
The rarefaction wave is a centered simple wave (see Fig. 4.54) and each characteristic is a straight line passing through the point (x R , t R ) and along which u and c are
constant. The point of intersection (x 1 , t 1 ) of the shock with the head or front of the
expansion wave becomes the origin of a new coordinate system x, t
ð Þ where x ¼
x À x R and t ¼ t À t 1 . It can be seen from Fig. 4.54 that each of the straight line
characteristics intersect the x-axis at a point on this axis which is denoted by the
symbol ξ and a typical characteristic is given by the equation,
x ¼ u þ c
ð
Þt þ ξ
ð4:65Þ
Each value of ξ denotes a specific straight line characteristic as shown in Fig. 4.54
and, hence, u and c are functions of ξ, so that the previous equation for a typical
characteristic can be written as
x ξ
ð Þ ¼ u ξ
ð Þ þ c ξ
ð Þ
½
Š t ξ
ð Þ þ ξ
ð4:66Þ
The intercept, ξ, on the x axis for the various rarefaction waves corresponds to the
following range;
c 0 t 1 À t R
ð
Þ< ξ
c 0 þ
γ þ 1
2
u p
h
i
t 1 À t R
ð
Þ
Fig. 4.54 New coordinate system for determining the shock decay process (see text)
4.8 Numerical Examples of Plane Shocks
195