Environmental Engineering Reference
In-Depth Information
the reflected waves always have opposite phase. Such a phenomenon is called
semi-
wave loss
. The forward, the backward and the reflected waves travel simultaneously
with a common velocity
a
. The effect of reflected waves arrives exactly at the instant
when the effect of forward waves disappears, regardless of the point
x
.
A better understanding of these phenomena can be achieved through drawing the
wave shape at some instant
t
0
for the case
ψ
(
x
)
≡
0 and for some typical wave shape
ϕ
(
x
)
such as triangle wave in Fig. 2.6.
Even Continuation
Find the solution of PDS
⎧
⎨
⎩
a
2
u
xx
,
u
tt
=
0
<
x
<
+
∞
,
0
<
t
,
u
x
(
0
,
t
)=
0
,
(2.70)
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
,
(2.71)
u
(
x
,
0
)=
Φ
(
x
)
,
u
t
(
x
,
0
)=
Ψ
(
x
)
,
where
Φ
(
x
)
and
Ψ
(
x
)
come from an even prolongation of
ϕ
(
x
)
and
ψ
(
x
)
, respec-
tively,
ϕ
(
x
)
,
x
>
0
,
ψ
(
x
)
,
x
>
0
,
Φ
(
x
)=
Ψ
(
x
)=
ϕ
(
−
x
)
,
x
≤
0
,
ψ
(
−
x
)
,
x
≤
0
.
The solution of PDS (2.71) can be expressed by the D'Alembert formula. It is
straightforward to show that the D'Alembert solution of PDS (2.71) satisfies
Fig. 2.6
Triangle wave
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