Biomedical Engineering Reference
In-Depth Information
Proposition 3.3.
The
λ
1
-and
λ
3
- characteristic fields are genuinely non-linear if the
tube law exponent
m
=
−
2andthe
λ
2
- characteistic field is linearly degenerate.
Proof.
Since
λ
2
-
characteristic field is linearly degenerate as claimed. For the other two characteristic
fields, some algebraic manipulations give
λ
2
=
0 it follows that
∇λ
2
=
0
and thus
∇λ
2
·
R
2
=
0. Therefore the
K
A
0
m
K
A
0
m
(
+
)
(
+
)
m
m
2
m
m
2
∇λ
1
·
R
1
=
2
A
m
,
∇λ
3
·
R
3
=
−
2
A
m
.
K
A
0
m
K
A
0
m
ρ
ρ
(
)
(
)
Therefore the
λ
Q
-and
λ
Q
- characteristic fields are genuinely non-linear pro-
1
3
=
−
vided
m
2, and the proof is complete.
2.3.3 Generalized Riemann invariants
The generalized Riemann invariants are relations that are valid across simple waves.
These are most conveniently expressed as a set of ordinary differential equations in
phase space, see [11] for details.
T
,
[
,
,...,
]
Proposition 3.4.
For a given hyperbolic system of
M
unknowns
w
1
w
2
w
M
T
the gen-
for any
λ
i
-characteristic field with right eigenvector
R
i
=[
r
1
i
,
r
2
i
,...,
r
Mi
]
eralized Riemann invariants are solutions of the following
M
−
1 ordinary differen-
tial equations in phase space
dw
1
r
1
i
=
dw
2
r
2
i
=
...
dw
n
r
Mi
.
(2.20)
Proof.
(omitted). See [11].
Proposition 3.5.
The generalized Riemann invariants for
λ
1
=
u
−
c
are
2
m
c
K
=
constant
,
+
u
=
constant
.
(2.21)
Proof.
Application of (2.20) from Proposition 3.4 to
λ
=
λ
1
=
u
−
c
with
R
1
=
T
, with
γ
1
[
1
,
u
−
c
,
0
]
γ
1
=
1 gives
dA
1
=
d
(
Au
)
dK
0
.
c
=
(2.22)
u
−
The last member of the two equalities gives the first sought result
K
constant
,as
desired, while the second result is obtained by manipulating the first equality, leading
to
=
c
(
A
)
dA
+
du
=
0
.
A
