Geoscience Reference
In-Depth Information
Performing the differentiations and cancelling sin
ϑ
,weget
(
u
2
+
E
2
)
∂
2
V
∂u
2
∂u
+
∂
2
V
1
u
2
+
E
2
cos
2
ϑ
+2
u
∂V
∆
V
≡
+
∂ϑ
2
(1-158)
∂λ
2
=0
,
u
2
+
E
2
cos
2
ϑ
(
u
2
+
E
2
)sin
2
ϑ
∂
2
V
cot
ϑ
∂V
∂ϑ
+
which is
Laplace's equation in ellipsoidal-harmonic coordinates
.Analterna-
tive expression is obtained by omitting the factor (
u
2
+
E
2
cos
2
ϑ
)
−
1
:
(
u
2
+
E
2
)
∂
2
V
∂u
2
∂u
+
∂
2
V
u
2
+
E
2
cos
2
ϑ
(
u
2
+
E
2
)sin
2
ϑ
∂
2
V
∂λ
2
+2
u
∂V
+cot
ϑ
∂V
∂ϑ
+
=0
.
∂ϑ
2
(1-159)
In the limiting case,
E
→
0, these equations reduce to the spherical expres-
sions (1-35) and (1-36).
1.16
Ellipsoidal harmonics
To solve (1-158) or (1-159), we proceed in a way which is analogous to
the method used to solve the corresponding equation (1-36) in spherical
coordinates. What we did there may be summarized as follows. By the trial
substitution
V
(
r, ϑ, λ
)=
f
(
r
)
g
(
ϑ
)
h
(
λ
)
,
(1-160)
we separated the variables
r, ϑ, λ
, so that the original partial differential
equation (1-36) was decomposed into three ordinary differential equations
(1-39), (1-46), and (1-47).
In order to solve Laplace's equation in ellipsoidal coordinates (1-159),
we correspondingly make the ansatz (trial substitution)
V
(
u, ϑ, λ
)=
f
(
u
)
g
(
ϑ
)
h
(
λ
)
.
(1-161)
Substituting and dividing by
fgh
,weget
u
2
+
E
2
cos
2
ϑ
(
u
2
+
E
2
)sin
2
ϑ
h
h
1
f
[(
u
2
+
E
2
)
f
+2
uf
]+
1
g
(
g
+
g
cot
ϑ
)+
=0
.
(1-162)
The variable
λ
occurs only through the quotient
h
/h
, which consequently
must be constant. One sees this more clearly by writing the equation in the
form
1
g
(
g
+
g
cot
ϑ
)
=
h
(
u
2
+
E
2
)sin
2
ϑ
u
2
+
E
2
cos
2
ϑ
f
[(
u
2
+
E
2
)
f
+2
uf
]+
1
−
.
h
(1-163)
