Environmental Engineering Reference
In-Depth Information
Proof.
For any point
M
on
S
,
the integrand in the single-layer potential
satisfies condition (7.148) in Theorem 2. Thus the single-layer
=
μ
(
P
)
F
(
M
,
P
)
r
PM
potential converges uniformly at all points
M
S
.
Therefore the single-layer potential exists at all points
M
in space, is a continuous
function of
M
, and is a harmonic function outside
S
provided that
S
is smooth,
bounded and with a continuous density function
∈
μ
(
P
)
.
Theorem 6.
Let
S
be a
in the double-
layer potential is continuous, the potential (Eq. (7.143)) is convergent for all
M
Ляпунов
surface. If the density function
τ
(
P
)
∈
S
.
Proof.
The integrand
τ
(
P
)
cos
(
PM
,
n
)
in the double-layer potential satisfies, by
r
PM
Theorem 3, the condition in Remark 2 of Theorem 2. Therefore the double-layer
potential
u
S
.
However, the double-layer potential
u
(
M
)
is convergent for
M
∈
(
)
M
is not continuous on
S
in general.
Ляпунов
τ
(
)
Theorem 7.
Let
S
be a closed
surface. Suppose that density function
P
is continuous on
S
. The double-layer potential
u
(
M
)
in Eq. (7.143) has a disconti-
nuity of the first kind at
M
∈
S
,i.e.
u
(
M
)=
u
(
M
)
−
2
πτ
(
M
)
,
M
∈
S
(7.151)
u
(
M
)=
u
(
M
)+
2
πτ
(
M
)
,
where
u
(
M
)
is the limit of
u
(
M
)
as
M
tends to
S
from the inside and
u
(
M
)
is the
limit of
u
(
M
)
as
M
tends to
S
from the outside.
Proof
. First we consider the case
)=
τ
0
(constant). Note that the solid angle of
area element d
S
at
P
on
S
viewing from
M
is
τ
(
P
cos
(
MP
,
n
)
cos
(
PM
,
n
)
d
ω
=
d
S
=
−
d
S
.
r
PM
r
PM
The double-layer potential is thus
⎧
⎨
−
4
πτ
0
,
M
∈
Ω
,
τ
0
cos
(
PM
,
n
)
u
1
(
M
)=
d
S
=
−
2
πτ
,
M
∈
S
,
0
r
PM
⎩
∈
Ω
.
S
0
,
M
Ω
are the domains inside
S
and outside
S
, respectively. For a point
P
on
S
,wehave
where
Ω
and
⎧
⎨
u
1
(
P
)=
−
4
πτ
0
,
u
1
(
P
)=
−
2
πτ
0
,
P
∈
S
,
(7.152)
⎩
u
1
(
P
)=
0
.
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