applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemati ...

, the Hough functions are the

eigenfunctions In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...

of Laplace's tidal equations which govern fluid motion on a rotating sphere. As such, they are relevant in

geophysics Geophysics () is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. The term ''geophysics'' so ...

and

meteorology Meteorology is a branch of the atmospheric sciences (which include atmospheric chemistry and physics) with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did no ...

where they form part of the solutions for atmospheric and ocean waves. These functions are named in honour of Sydney Samuel Hough.Hough, S. S. (1898)
On the application of harmonic analysis to the dynamical theory of the tides. Part II. On the general integration of Laplace's dynamical equations
Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, vol. 191, 139–185. Each Hough mode is a function of

latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...

and may be expressed as an infinite sum of associated Legendre polynomials; the functions are orthogonal over the sphere in the continuous case. Thus they can also be thought of as a

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. Here we consider that of square-integrable functions de ...

in which the basis functions are the normal modes of an atmosphere at rest.

See also

References

Further reading