>> What are spherical harmonics

The spherical harmonics are the angular solution to Laplace's equation in spherical coordinates.

In other words: Laplace's equation describes situations in which most kinds of oscillations occur. In Cartesian coordinates the obvious possible solution are plain waves . In nature, whenever you encounter waves it's probably because the local interaction between particles is described by laplace's equation. This is true for acoustic waves, electromagnetic waves, surface waves, hosepipe waves and to a lesser degree waves in liquids. Spherical harmonics describe these waves when they propagate on the surface of a sphere.

Consult you favorite math textbook to learn more about spherical harmonics!

>> The equation


where

Note: is the polar coordinate with , is the azimuthal (longitudinal) coordinate with . This is the convention normally used in physics, but differs from the one usually found in mathematical literature.

>> The animation (click to download mpeg file)

l=1
m=0

m=1
l=2
m=0

m=1

m=2
l=3
m=0

m=1

m=2

m=3
l=4
m=0

m=1

m=2

m=3

m=4