Environmental Engineering Reference
In-Depth Information
reads, by the solution structure theorem,
⎡
⎤
t
a
2
t
⎣
M
,
τ
)
t
⎦
1
f
(
(
,
)=
W
f
τ
(
,
−
τ
)
τ
=
u
M
t
M
t
d
d
S
d
τ
4
π
−
τ
0
0
S
a
(
t
−
τ
)
⎡
⎤
a
2
at
r
a
)
M
,
1
f
(
t
−
⎣
⎦
=
d
S
d
r
4
π
r
0
S
r
a
2
V
at
M
,
r
f
(
t
−
a
)
1
=
d
v
,
(2.85)
4
π
r
where
V
at
stands for a sphere of center
M
and radius
at
. Equation (2.85) is called
the
Kirchhoff formula of three-dimensional wave equations
.
2.8.2 Method of Spherical Means
Consider spherical surface
S
M
r
of a sphere of center
M
0
and radius
r
. The averaged
value of
u
on
S
M
0
(
M
,
t
)
is
r
r
2
S
M
0
r
1
u
(
r
,
t
)=
u
(
M
,
t
)
d
S
.
4
π
Also
u
(
M
0
,
t
0
)=
r
→
0
u
lim
(
r
,
t
0
)
.
Once
u
(
r
,
t
0
)
is available, therefore, we can obtain
u
(
M
0
,
t
0
)
simply by taking the
0. This is the essence of the method of spherical means. The
method may be very useful for finding solutions of wave equations.
Let
V
M
0
limit of
u
(
r
,
t
0
)
as
r
→
be a sphere of center
M
0
and radius
r
. Integrating a wave equation over
r
V
M
0
yields
r
2
u
∂
a
2
t
2
d
v
=
Δ
u
d
v
.
∂
V
M
0
r
V
M
0
r
Search WWH ::
Custom Search