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In applied mathematics, the Hough functions are the eigenfunctions of Laplace's tidal equations which govern fluid motion on a rotating sphere. As such, they are relevant in geophysics and meteorology where they form part of the solutions for tide. These functions are named in honour of Sydney Samuel Hough.Hough, S. S. (1897). 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.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 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 are the of an atmosphere at rest.
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