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{{Short description|Eigenfunctions of Laplace's tidal equations which govern fluid motion on a rotating sphere}}
In [[applied mathematics]], the '''Hough functions''' are the [[eigenfunctions]] of [[Primitive equations|Laplace's tidal equations]] which govern [[secondary circulation|fluid motion on a rotating sphere]]. As such, they are relevant in [[geophysics]] and [[meteorology]] where they form part of the solutions for [[tide|atmospheric and ocean waves]]. These functions are named in honour of [[Sydney Samuel Hough]].<ref>{{cite book|author=Cartwright, David Edgar|title=Tides: A Scientific History|year=2000|publisher=Cambridge University Press|pages=[https://archive.org/details/tidesscientifich0000cart/page/85 85]–87|isbn=9780521621458 |url=https://archive.org/details/tidesscientifich0000cart|url-access=registration}}</ref><ref>Hough, S. S. (1897). [https://babel.hathitrust.org/cgi/pt?id=coo.31924060893561;view=1up;seq=259 On the Application of Harmonic Analysis to the Dynamical Theory of the Tides. Part I. On Laplace's' Oscillations of the First Species, and on the Dynamics of Ocean Currents]. Proceedings of the Royal Society of London, vol. 61, 201–257.</ref><ref>Hough, S. S. (1898). [https://babel.hathitrust.org/cgi/pt?id=coo.31924060893355;view=1up;seq=203 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.</ref>

Each Hough mode is a function of [[latitude]] 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 which the [[basis function]]s are the [[normal mode]]s of an atmosphere at rest.

==See also==
*[[Secondary circulation]]
*[[Legendre polynomials]]
*[[Primitive equations]]

==References==
{{reflist}}

== Further reading ==

* {{cite journal
|author=Lindzen, R.S.
|year=2003
|title=The Interaction of Waves and Convection in the Tropics
|journal=Journal of the Atmospheric Sciences
|volume=60
|issue=24
|pages=3009–3020
|url=http://eaps.mit.edu/faculty/lindzen/Waves_and_Convection031.pdf
|bibcode=2003JAtS...60.3009L
|doi=10.1175/1520-0469(2003)060<3009:TIOWAC>2.0.CO;2
|access-date=2009-03-22
|archive-date=2010-06-13
|archive-url=https://web.archive.org/web/20100613123012/http://eaps.mit.edu/faculty/lindzen/Waves_and_Convection031.pdf
|url-status=dead
}}

[[Category:Atmospheric dynamics]]
[[Category:Physical oceanography]]
[[Category:Fluid mechanics]]
[[Category:Special functions]]

Hough function - Revision history

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