The pressure, p 2 , behind the advancing shock wave in terms of the diaphragm
pressure ratio, p 4 /p 1 , is given by Eq. (4.57) and it is straightforward to show, as
previously established in Sect. 4.8.7, that p 2 ¼ 2.848, which is in good agreement
with the numerical output as shown in Fig. B.2. Once the shock wave reflects at the
end wall located at x ¼ 10, the magnitude of the reflected pressure, p 3 , is given by
Eq. (3.47), namely,
p 3
p 2
¼
3γ À 1
ð
Þp 2 À γ À 1
ð
Þp 1
γ À 1
ð
Þp 2 þ γ þ 1
ð
Þp 1
,
hence, we find that p 3 ¼ 7.012 which is in general agreement with the numerically
calculated value as shown in Fig. B.3. The variation of these parameters at much
later times as a function of particle position are shown in Figs. B.4, B.5, and B.6 and
the markers in all plots show the position of the contact surface at the times indicated.
Fig. B.2 Plots of pressure, particle velocity and density as a function of particle position at
t ¼ 400Δt. For the numerical calculations the following parameters apply; γ ¼ 1.4, κ ¼ 1.2,
Δx ¼ 0.1 and Δt ¼ 0.005 (see text)
334
Appendix B