In applied mathematics

Applied mathematics

Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...

, the Hough functions are the eigenfunctions of Laplace's tidal equations

Primitive equations

The primitive equations are a set of nonlinear differential equations that are used to approximate global atmospheric flow and are used in most atmospheric models...

 which govern fluid motion on a rotating sphere

Secondary circulation

A secondary circulation is a circulation induced in a rotating system. For example, the primary circulation of Earth's atmosphere is zonal. If however a parcel of air, that moves in a purely zonal direction, is accelerated or decelerated zonally, the Coriolis force will add a meridional component...

. As such, they are relevant in geophysics

Geophysics

Geophysics is the physics of the Earth and its environment in space; also the study of the Earth using quantitative physical methods. The term geophysics sometimes refers only to the geological applications: Earth's shape; its gravitational and magnetic fields; its internal structure and...

 and meteorology

Meteorology

Meteorology is the interdisciplinary scientific study of the atmosphere. Studies in the field stretch back millennia, though significant progress in meteorology did not occur until the 18th century. The 19th century saw breakthroughs occur after observing networks developed across several countries...

 where they form part of the solutions for atmospheric and ocean waves

Tide

Tides are the rise and fall of sea levels caused by the combined effects of the gravitational forces exerted by the moon and the sun and the rotation of the Earth....

.

Each Hough mode is a function of latitude

Latitude

In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

 and may be expressed as an infinite sum of associated Legendre polynomials

Associated Legendre polynomials

In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation\,y -2xy' + \left\,y = 0,\,or equivalently...

; the functions are orthogonal over the sphere in the continuous case. Thus they can also be thought of as a generalized Fourier series

Generalized Fourier series

In mathematical analysis, many generalizations of Fourier series have proved to be useful.They are all special cases of decompositions over an orthonormal basis of an inner product space....

 in which the basis function

Basis function

In mathematics, a basis function is an element of a particular basis for a function space. Every continuous function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis...

s are the normal mode

Normal mode

A normal mode of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The frequencies of the normal modes of a system are known as its natural frequencies or resonant frequencies...

s of an atmosphere at rest.