Environmental Engineering Reference
In-Depth Information
The key is to perform the integration over
V
at
first over a spherical surface and then
over the radius. The integration over the spherical surface can be reduced to one
over
D
at
. The result is
⎡
⎣
⎤
⎦
t
M
,
τ
)
1
f
(
u
(
x
,
y
,
t
)=
d
S
d
τ
4
π
a
a
(
t
−
τ
)
0
S
a
(
t
−
τ
)
⎡
⎣
⎤
⎦
t
1
f
(
ξ
,
η
,
τ
)
a
2
=
d
ξ
d
η
d
τ
,
(2.95)
2
π
a
0
2
2
2
(
t
−
τ
)
−
(
ξ
−
x
)
−
(
η
−
y
)
D
a
(
t
−
τ
)
which is called the
Kirchhoff formula of two-dimensional wave equations.
If
u
(
x
,
y
,
t
)
represents the displacement,
⎡
⎣
⎤
⎦
[
1
a
−
1
[
u
]=[
][
f
]
a
2
d
ξ
d
η
d
τ
]
2
2
2
(
t
−
τ
)
−
(
ξ
−
x
)
−
(
η
−
y
)
T
L
·
1
T
2
·
1
L
·
L
2
T
=
=
L
,
which shows the unit correctness in Eq. (2.95). Such a unit check can help us to find
some errors that may arise in derivation.
Remark 4.
By the principle of superposition, the solution of
u
tt
=
a
2
Δ
u
+
f
(
x
,
y
,
t
)
, −
∞
<
x
,
y
<
+
∞
,
0
<
t
,
u
(
x
,
y
,
0
)=
ϕ
(
x
,
y
)
,
u
t
(
x
,
y
,
0
)=
ψ
(
x
,
y
)
can be obtained by Eqs. (2.93) and (2.95),
⎡
⎣
∂
∂
1
ϕ
(
ξ
,
η
)
(
,
)=
u
M
t
d
ξ
d
η
2
π
a
t
2
2
2
(
at
)
−
(
ξ
−
x
)
−
(
η
−
y
)
D
at
⎤
ψ
(
ξ
,
η
)
⎦
+
d
ξ
d
η
2
2
2
(
at
)
−
(
ξ
−
x
)
−
(
η
−
y
)
D
at
⎡
⎤
t
⎣
⎦
1
f
(
ξ
,
η
,
τ
)
+
a
2
d
ξ
d
η
d
τ
.
2
π
a
0
2
2
2
(
−
τ
)
−
(
ξ
−
)
−
(
η
−
)
t
x
y
D
a
(
t
−
τ
)
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