We have already established that M i ¼ 1.164 and γ ¼ 1.4, and as all quantities are
known on the right-hand side of the above equation we obtain, after substitution, the
following quadratic equation for M r ,
3:448M
2
r À M r À 3:448 ¼ 0:
Solving this quadratic, we find (ignoring the negative root) that M r ¼ 1.155 which
is the Mach number of the reflected shock relative to the air into which it is moving.
We have already shown from Eqs. (3.25) and (3.28) that p 2 ¼ 1.414 and ρ 2 ¼ 1.279,
so that the sonic velocity in the air that has been traversed by the incident shock is
given by
c 2 ¼
ffiffiffiffiffiffiffi
γp 2
ρ 2
r
,
hence,
c 2 ¼ 1:244:
From Sect. 3.11 we have the relationship for the velocity of the reflected shock,
U r , in terms of the Mach number, M r , the piston velocity u p and sound speed c 2
according to the equation,
M r ¼
U r þ u p
c 2
:
Substituting the known quantities in this latter equation, we find that
U r ¼ 1:137:
Referring now to the numerical results and let us take, for example, the pressure
plot shown in Fig. 4.22. We find that the position of the reflected shock
corresponding to the plots of p 1500, j and p 1800, j occur at x 1500, 122 ¼ 60.65 and at
x 1800, 53 ¼ 43.57, respectively. The specific values for j were determined by locating
the position where the artificial viscosity q attains its maximum value in each case.
Since the time difference between these maxima is (1800 À 1500)Δt ¼ 15, the
velocity of the reflected shock wave is given by
U r ¼
60:65 À 43:57
15
¼ 1:138,
which is in very good agreement with the value above.
4.8 Numerical Examples of Plane Shocks
167
known on the right-hand side of the above equation we obtain, after substitution, the
following quadratic equation for M r ,
3:448M
2
r À M r À 3:448 ¼ 0:
Solving this quadratic, we find (ignoring the negative root) that M r ¼ 1.155 which
is the Mach number of the reflected shock relative to the air into which it is moving.
We have already shown from Eqs. (3.25) and (3.28) that p 2 ¼ 1.414 and ρ 2 ¼ 1.279,
so that the sonic velocity in the air that has been traversed by the incident shock is
given by
c 2 ¼
ffiffiffiffiffiffiffi
γp 2
ρ 2
r
,
hence,
c 2 ¼ 1:244:
From Sect. 3.11 we have the relationship for the velocity of the reflected shock,
U r , in terms of the Mach number, M r , the piston velocity u p and sound speed c 2
according to the equation,
M r ¼
U r þ u p
c 2
:
Substituting the known quantities in this latter equation, we find that
U r ¼ 1:137:
Referring now to the numerical results and let us take, for example, the pressure
plot shown in Fig. 4.22. We find that the position of the reflected shock
corresponding to the plots of p 1500, j and p 1800, j occur at x 1500, 122 ¼ 60.65 and at
x 1800, 53 ¼ 43.57, respectively. The specific values for j were determined by locating
the position where the artificial viscosity q attains its maximum value in each case.
Since the time difference between these maxima is (1800 À 1500)Δt ¼ 15, the
velocity of the reflected shock wave is given by
U r ¼
60:65 À 43:57
15
¼ 1:138,
which is in very good agreement with the value above.
4.8 Numerical Examples of Plane Shocks
167