c
0 u x 0 , 0
ð
Þ
ð
޼2
ffiffi ffi
2
p e
À
1 ffi ffi
2
p
2
¼ À0:858
so that
t B ¼ 1:165:
Accordingly, the earliest time for the solution to become multi-valued is 1.165
and the profile develops a vertical slope at x ¼
ffiffi ffi
2
p
.
2.7 Application of Riemann Invariants to Simple Flow
Problems
Returning to the Riemann invariants we found that u + 2c/(γ À 1) is a constant on the
positive characteristic C + given by dx/dt ¼ u + c and u À 2c/(γ À 1) is constant on the
negative characteristic C À given by dx/dt ¼ u À c. Let
R þ ¼ u þ
2c
γ À 1
ð2:61Þ
and
R À ¼ u À
2c
γ À 1
:
ð2:62Þ
Solving these equations for u and c yields,
u ¼
R þ þ R À
2
and c ¼
γ À 1
ð
Þ
4
R þ À R À
ð
Þ :
Hence, on C + we obtain,
dx
dt
¼
γ þ 1
4
R þ þ
3 À γ
4
R À
ð2:63Þ
and on C À we obtain,
dx
dt
¼
3 À γ
4
R þ þ
γ þ 1
4
R À :
ð2:64Þ
2.7 Application of Riemann Invariants to Simple Flow Problems
73