Environmental Engineering Reference
In-Depth Information
respectively,
⎧
⎨
⎧
⎨
−
ϕ
(
−
)
,
<
,
−
ψ
(
−
)
,
<
,
x
x
0
x
x
0
Φ
(
)=
0
,
x
=
0
,
Ψ
(
)=
0
,
x
=
0
,
x
x
⎩
⎩
ϕ
(
x
)
,
x
>
0
,
ψ
(
x
)
,
x
>
0
.
Clearly, PDS (2.68) is exactly the same as PDS (2.67) in 0
<
x
<
+
∞
,0
<
t
.Bythe
D'Alembert formula and focusing on the region of
x
≥
0 , we obtain the solution of
PDS (2.67)
at
x
−
at
Ψ
(
ξ
)
x
+
)=
Φ
(
x
+
at
)+
Φ
(
x
−
at
)
1
2
a
(
,
+
u
x
t
d
ξ
2
⎧
⎨
ϕ
(
+
)+
ϕ
(
−
)
x
+
at
x
at
x
at
1
2
a
x
a
≥
+
ψ
(
ξ
)
d
ξ
,
t
,
2
x
−
at
=
(2.69)
⎩
at
at
−
x
ψ
(
ξ
)
x
+
ϕ
(
x
+
at
)
−
ϕ
(
at
−
x
)
1
2
a
x
a
<
+
d
ξ
,
t
,
2
in which we have used, for
x
<
at
,
x
+
at
x
−
at
Ψ
(
ξ
)
0
x
−
at
Ψ
(
ξ
)
x
+
at
d
ξ
=
d
ξ
+
Ψ
(
ξ
)
d
ξ
0
0
x
−
at
ψ
(
−
μ
)
x
+
at
=
−
d
μ
+
ψ
(
ξ
)
d
ξ
0
x
+
at
=
x
ψ
(
ξ
)
d
ξ
.
at
−
The wave shape of PDS (2.67) at
t
0
can be drawn by two approaches: using
the part of the wave shape of PDS (2.68) in
and drawing directly based on
Eq. (2.69). Although the detailed procedure differs for the two approaches, the final
result is the same. It is helpful when using the latter to understand the meaning of
terms in Eq. (2.69). When
t
[
0
,
+
∞
]
x
a
, the effect of the end
x
0 has not propagated
to the point
x
yet, so the D'Alembert formula is valid and the superposition is that
of initial forward and backward waves. When
t
<
=
x
a
, however, the effect of the
>
0 has already propagated to the point
x
. The forward waves
ϕ
(
x
−
at
)
end
x
=
and
2
0
1
2
a
should thus be replaced by the reflected waves
−
ϕ
(
at
−
x
)
at
ψ
(
ξ
)
d
ξ
and
2
x
−
0
at
−
x
ψ
(
ξ
)
1
2
a
d
ξ
, respectively. The reflected waves are also forward waves. For the
end
x
=
0, the backward wave is also called an
incoming wave
. The incoming and
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