Geoscience Reference
In-Depth Information
where the functions
g
and
h
individually depend on one variable only. Per-
forming this substitution in (1-40) and multiplying by sin
2
ϑ/g h
, we find
sin
ϑg
+cos
ϑg
+
n
(
n
+1)sin
ϑg
=
h
h
sin
ϑ
g
−
,
(1-45)
where the primes denote differentiation with respect to the argument:
ϑ
in
g
and
λ
in
h
. The left-hand side is a function of
ϑ
only, and the right-hand
side is a function of
λ
only. Therefore, both sides must again be constant;
let the constant be
m
2
. Thus, the partial differential equation (1-40) splits
into two ordinary differential equations for the functions
g
(
ϑ
)and
h
(
λ
):
sin
ϑg
(
ϑ
)+cos
ϑg
(
ϑ
)+
n
(
n
+1)sin
ϑ −
g
(
ϑ
) = 0 ;
m
2
sin
ϑ
(1-46)
h
(
λ
)+
m
2
h
(
λ
)=0
.
(1-47)
Solutions of Eq. (1-47) are the functions
h
(
λ
)=cos
mλ
and
h
(
λ
)=sin
mλ ,
(1-48)
as may be verified by substitution. Equation (1-46), Legendre's differential
equation, is more dicult. It can be shown that it has physically meaningful
solutions only if
n
and
m
are integers 0
,
1
,
2
,...
and if
m
is smaller than or
equal to
n
. A solution of (1-46) is the Legendre function
P
nm
(cos
ϑ
), which
will be considered in some detail in the next section. Therefore,
g
(
ϑ
)=
P
nm
(cos
ϑ
)
(1-49)
and the functions
Y
n
(
ϑ, λ
)=
P
nm
(cos
ϑ
)cos
mλ
and
Y
n
(
ϑ, λ
)=
P
nm
(cos
ϑ
)sin
mλ
(1-50)
are solutions of the differential equation (1-40) for Laplace's surface spherical
harmonics.
Since these solutions are linear, any linear combination of the solutions
(1-50) is also a solution. Such a linear combination has the general form
Y
n
(
ϑ, λ
)=
n
m
=0
[
a
nm
P
nm
(cos
ϑ
)cos
mλ
+
b
nm
P
nm
(cos
ϑ
)sin
mλ
]
,
(1-51)
where
a
nm
and
b
nm
are arbitrary constants. This is the general expression
for the surface spherical harmonics
Y
n
(
ϑ, λ
).
