Geoscience Reference
In-Depth Information
S
.Inthe
third boundary-value problem
, a linear combination of
V
and of its
normal derivative
hV
+
k
∂V
∂n
(1-125)
is given on
S
.
For the sphere, the solution of these boundary-value problems is also
easily expressed in terms of spherical harmonics. We consider the exterior
problems only, because these are of special interest to geodesy.
In
Neumann's problem
, we expand the given values of
∂V/∂n
on the
sphere
r
=
R
into a series of surface spherical harmonics:
∂V
∂n
=
∞
Y
n
(
ϑ, λ
)
.
(1-126)
r
=
R
n
=0
The harmonic function which solves Neumann's problem for the exterior of
the sphere is then
R
r
n
+1
Y
n
(
ϑ, λ
)
n
+1
R
∞
V
e
(
r, ϑ, λ
)=
−
.
(1-127)
n
=0
To verify it, we differentiate with respect to
r
, getting
R
r
n
+2
=
∞
∂V
e
∂r
Y
n
(
ϑ, λ
)
.
(1-128)
n
=0
Since for the sphere the normal coincides with the radius vector, we have
∂V
∂n
=
∂V
∂r
,
(1-129)
r
=
R
r
=
R
and we see that (1-126) is satisfied.
The
third boundary-value problem
is particularly relevant to physical
geodesy, because the determination of the undulations of the geoid from
gravity anomalies is just such a problem. To solve the general case, we again
expand the function defined by the given boundary values into surface spher-
ical harmonics:
=
∞
hV
+
k
∂V
∂n
Y
n
(
ϑ, λ
)
.
(1-130)
n
=0
The harmonic function
V
e
(
r, ϑ, λ
)=
∞
R
r
n
+1
Y
n
(
ϑ, λ
)
(1-131)
h
−
(
k/R
)(
n
+1)
n
=0
