spherical harmonics

(harmonic functions on the surface of a sphere)

Spherical harmonics are functions on the surface of a sphere which fulfill the same role as the sine function providing harmonics for periodic functions.

Laplace spherical harmonics (a common type, often what is meant by spherical harmonics) effectively divide the sphere into portions, portioning the sphere by number of meridian divisions and number of latitude divisions. The harmonic's mode is designated by two numbers, l and m, l (the degree, multipole moment or multipole number) being a natural number indicating the number of latitude-like divisions, and m (the order or azimuthal number) being a natural number indicating the number of meridian-like divisions, there being no more meridian divisions than latitude divisions. For example, for l=1, m=1, the sphere is divided along an equator-like line and a meridian-like line, resulting in four portions.

Spherical harmonics are used for describing gravity fields of planets (and the letters l and m probably grew out of analysis Earth's gravity field in terms of latitude-divided and meridian-divided modes). They are also used in describing seismology (including asteroseismology), and are of interest in the theory of core collapse supernovae. They are also used in characterizing the distribution of the cosmic microwave background (CMB) variations around the celestial sphere (CMB anisotropies). They can be used in characterizing weather around a world. They are also used within a technique for solving some types of differential equations.

(mathematics)