Spherical harmonics (normalized) in the radial direction are expressed by the equation:
,
where
and
are the degree and order, respectively, of the generalized Legendre function
. The function is expressed on a sphere with polar angle
(starting from 0 at the positive
axis) and azimuth
(counterclockwise from the
axis in the equatorial plane as viewed from the positive
axis).
For display purposes in this Demonstration,
is multiplied by the time function
, scaled by a constant [0,1] and then added to 1.0. The radial motion in a plane given by constant
or
is then shown as a function of time, with a complete oscillation every
units of time.
There are three common classes of radial spherical harmonics. The first, zonal harmonics of degree
, is given when
. For this class, there is no longitudinal variation and there are zeros along
parallels of latitude. The second, sectoral harmonics of degree
and order
, has no latitudinal zeros but has zeros along
longitudes. The third, tesseral harmonics of degree
and order
(
), has zeros along
parallels of latitude and zeros along
longitudes.
![[Snapshot]](HTMLImages/index.en/thumbnail_1.gif)
![[Snapshot]](HTMLImages/index.en/thumbnail_2.gif)
![[Snapshot]](HTMLImages/index.en/thumbnail_3.gif)
Embed Interactive Demonstration 
Browse all topics
