Geoscience Reference
In-Depth Information
only on
ϑ
and
λ
, both sides must be constant. We can therefore separate the
equation into two equations:
r
2
f
(
r
)+2
rf
(
r
)
−
n
(
n
+1)
f
(
r
) = 0
(1-39)
and
∂
2
Y
∂ϑ
2
∂
2
Y
∂λ
2
+cot
ϑ
∂Y
1
sin
2
ϑ
∂ϑ
+
+
n
(
n
+1)
Y
=0
,
(1-40)
where we have denoted the constant by
n
(
n
+1).
Solutions of (1-39) are given by the functions
f
(
r
)=
r
n
and
f
(
r
)=
r
−
(
n
+1)
;
(1-41)
this should be verified by substitution. Denoting the still unknown solutions
of (1-40) by
Y
n
(
ϑ, λ
), we see that Laplace's equation (1-35) is solved by the
functions
V
=
r
n
Y
n
(
ϑ, λ
)and
V
=
Y
n
(
ϑ, λ
)
r
n
+1
.
(1-42)
These functions are called
solid spherical harmonics
, whereas the functions
Y
n
(
ϑ, λ
) are known as (Laplace's)
surface spherical harmonics
. Both kinds
are called
spherical harmonics
; the kind referred to can usually be judged
from the context.
Note that
n
is not an arbitrary constant but must be an integer 0
,
1
,
2
,...
as we will see later. If a differential equation is linear, and if we know several
solutions, then, as is well known, the sum of these solutions is also a solution
(this holds for
all
linear equation systems!). Hence, we conclude that
V
=
∞
r
n
Y
n
(
ϑ, λ
)and
V
=
∞
Y
n
(
ϑ, λ
)
r
n
+1
(1-43)
n
=0
n
=0
are also solutions of Laplace's equation ∆
V
= 0; that is, harmonic functions.
The important fact is that
every
harmonic function - with certain restrictions
- can be expressed in one of the forms (1-43).
1.6
Surface spherical harmonics
Now we have to determine Laplace's surface spherical harmonics
Y
n
(
ϑ, λ
).
We attempt to solve (1-40) by a new trial substitution
Y
n
(
ϑ, λ
)=
g
(
ϑ
)
h
(
λ
)
,
(1-44)
