Environmental Engineering Reference
In-Depth Information
is
a
2
at
rf
t
d
r
1
r
a
u
(
x
,
t
)=
−
.
0
Proof.
By Eq. (2.98),
t
x
+
a
(
t
−
τ
)
t
1
2
a
1
2
a
u
(
x
,
t
)=
d
τ
f
(
τ
)
d
ξ
=
f
(
τ
)
2
a
(
t
−
τ
)
d
τ
0
x
−
a
(
t
−
τ
)
0
a
2
at
rf
t
d
r
1
r
a
=
−
.
0
r
a
. Thus d
1
a
d
r
,and
r
varies from
Introduce a new variable
r
such that
τ
=
t
−
τ
=
−
at
to 0 when
τ
changes from 0 to
t
. Therefore
a
2
at
rf
t
d
r
1
r
a
u
(
x
,
t
)=
−
.
0
Example 3.
Find the solution of
u
tt
=
a
2
Δ
u
,
−
∞
<
x
,
y
,
z
<
+
∞
,
0
<
t
,
x
2
u
(
r
,
0
)=
ϕ
(
r
)
,
u
t
(
r
,
0
)=
ψ
(
r
)
,
r
=
+
y
2
+
z
2
.
Solution.
Since
ϕ
(
r
)
and
ψ
(
r
)
are functions of
r
only, they have spherical symmetry
so that
ϕ
(
−
r
)=
ϕ
(
r
)
and
ψ
(
−
r
)=
ψ
(
r
)
for a generalized polar coordinate system
in which points
are symmetric with respect to the origin.
Therefore the problem is a Cauchy problem in
(
−
r
,
θ
,
ϕ
)
and
(
r
,
θ
,
ϕ
)
−
∞
<
r
<
+
∞
,0
<
t
. Also, under a
spherical symmetry,
r
2
∂
r
2
∂
1
u
Δ
u
(
r
,
t
)=
, −
∞
<
r
<
+
∞
,
0
<
t
.
∂
r
∂
r
The problem becomes,
⎧
⎨
2
2
(
)
(
)
∂
ru
a
2
∂
ru
=
,−
∞
<
r
<
+
∞
,
0
<
t
,
∂
t
2
∂
r
2
⎩
ru
(
r
,
0
)=
r
ϕ
(
r
)
,
ru
t
(
r
,
0
)=
r
ψ
(
r
)
.
By the D'Alembert formula (2.97) we obtain the solution
)=
(
r
+
at
)
ϕ
(
r
+
at
)+(
r
−
at
)
ϕ
(
r
−
at
)
u
(
r
,
t
2
r
r
+
at
1
2
ar
+
ξψ
(
ξ
)
ξ
.
d
r
−
at
Remark 5.
It is always useful to study a problem by using different methods. We
have arrived at Eq. (2.98), the Kirchhoff formula of one-dimensional wave equa-
tions, using the method of descent and the solution structure theorem that shares the
Search WWH ::
Custom Search