Environmental Engineering Reference
In-Depth Information
Adding these two equations yields
x
2
x
2
(
)+
(
)=
ϕ
+
ψ
−
[
(
)+
(
)]
f
x
g
x
f
0
g
0
and
f
(
0
)+
g
(
0
)=
ϕ
(
0
)
.
Finally, we obtain
x
x
+
at
−
at
(
,
)=
ϕ
+
ψ
−
ϕ
(
)
.
u
x
t
0
2
2
2.7.5 Problems in a Semi-Infinite Domain
and the Method of Continuation
Problems in a semi-infinite domain requires solving wave equations in the region:
0
t
. We discuss the method of continuation for solving these prob-
lems by using homogeneous wave equations as an example. For problems in a semi-
infinite domain, we require the boundary condition
x
<
x
<
+
∞
,0
<
0 in addition to the initial
conditions. Note that if an odd function of
x
has the definition at
x
=
0, its value must
be zero. Similarly, the derivative of an even function of
x
must be vanished at
x
=
=
0
if it is differentiable at
x
=
0. We should make an odd continuation when
u
(
0
,
t
)=
0,
and an even continuation when
u
x
(
0. Such a continuation of initial conditions
does not change the CDS of the problem and reduces the problem into an auxiliary
problem in an infinite domain which can be solved by the D'Alembert formula. The
solution of the original problem in a semi-infinite domain can be obtained by using
the solution of the auxiliary problem in 0
0
,
t
)=
<
x
<
+
∞
,0
<
t
.
Odd Continuation
Find the Solution of PDS
⎧
⎨
a
2
u
xx
,
u
tt
=
0
<
x
<
+
∞
,
0
<
t
,
u
(
0
,
t
)=
0
,
(2.67)
⎩
u
(
x
,
0
)=
ϕ
(
x
)
,
u
t
(
x
,
0
)=
ψ
(
x
)
.
Solution.
Consider an auxiliary problem
u
tt
=
a
2
u
xx
, −
∞
<
x
<
+
∞
,
0
<
t
,
(2.68)
u
(
x
,
0
)=
Φ
(
x
)
,
u
t
(
x
,
0
)=
Ψ
(
x
)
,
where
Φ
(
x
)
and
Ψ
(
x
)
come from an odd continuation of
ϕ
(
x
)
and
ψ
(
x
)
,
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