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.