Spherical Harmonics
>> 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
|