Environmental Engineering Reference
In-Depth Information
2.7.2 Method of Characteristics
a
2
u
xx
are (Sect. 1.1.5)
The characteristic curves of the wave equation
u
tt
=
x
+
at
=
c
1
,
x
−
at
=
c
2
.
A variable transformation
ξ
=
x
+
at
,
η
=
x
−
at
a
2
u
xx
into
u
ξη
=
reduces
u
tt
=
0. A twice integration with respect to
ξ
and
η
,
respectively, leads to
u
=
f
(
ξ
)+
g
(
η
)=
f
(
x
+
at
)+
g
(
x
−
at
)
(2.64)
(
,
)=
where
f
and
g
are differentiable functions. Applying initial conditions
u
x
0
0
(
,
)=
ψ
(
)
and
u
t
x
0
x
yields
⎧
⎨
f
(
x
)+
g
(
x
)=
0
,
1
a
ψ
(
f
(
g
(
x
)
−
x
)=
x
)
,
or
x
x
0
ψ
(
ξ
)
⎩
1
a
f
(
x
)
−
g
(
x
)=
d
ξ
+
C
,
=
(
)
−
(
)
where
C
f
x
0
g
x
0
and
x
0
is an arbitrary point. Therefore
x
x
0
ψ
(
ξ
)
x
x
0
ψ
(
ξ
)
1
2
a
C
2
,
1
2
a
C
2
.
f
(
x
)=
d
ξ
+
g
(
x
)=
−
d
ξ
−
ϕ
=
=
Consequently, the solution of PDS (2.60) for the case of
f
0is
x
+
at
1
2
a
u
(
x
,
t
)=
W
ψ
(
x
,
t
)=
ψ
(
ξ
)
d
ξ
.
x
−
at
Finally, the solution structure theorem shows that the solution of PDS (2.60) is
Eq. (2.61).
a
2
u
xx
Note.
It can be shown that the
u
(
x
,
t
)
in Eq. (2.62) satisfies the equation
u
tt
=
C
2
C
1
ϕ
(
)
∈
(
−
∞
,
+
∞
)
ψ
(
)
∈
(
−
∞
,
+
∞
)
and the initial conditions
x
,
x
,soitisasolu-
ϕ
(
)
ψ
(
)
tion. The demand for the smoothness of
x
and
x
by Eq. (2.62) itself is quite
weak, and the D'Alembert formula still works for those
not satisfy-
ing the above conditions. Therefore, the solution in Eq. (2.62) is called a
generalized
solution
or a
weak solution
. The former [the one satisfying
ϕ
(
x
)
and
ψ
(
x
)
C
2
ϕ
(
x
)
∈
(
−
∞
,
+
∞
)
and
C
1
] is called the
classical solution
.
The solution (2.62) is also unique because it comes rigorously from the technique
of integral transformation and characteristic curves based on the initial conditions. It
is also stable with respect to initial conditions in the sense of uniformapproximation.
ψ
(
x
)
∈
(
−
∞
,
+
∞
)
Search WWH ::
Custom Search