# Wave surface

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{{distinguish|Wave front}}
In mathematics, Fresnel's '''wave surface''', found by [Augustin-Jean Fresnel](/source/Augustin-Jean_Fresnel) in 1822, is a [quartic surface](/source/quartic_surface) describing the [propagation of light](/source/propagation_of_light) in an [optically biaxial crystal](/source/optically_biaxial_crystal). Wave surfaces are special cases of [tetrahedroid](/source/tetrahedroid)s which are in turn special cases of  [Kummer surface](/source/Kummer_surface)s.

In projective coordinates (''w'':''x'':''y'':''z'') the wave surface is given by
:<math>
\frac{a^2x^2}{x^2+y^2+z^2-a^2w^2} + \frac{b^2y^2}{x^2+y^2+z^2-b^2w^2} + \frac{c^2z^2}{x^2+y^2+z^2-c^2w^2} =0
</math>
They are used in the treatment of [conical refractions](/source/Conical_refraction).thumb|Fresnel's Wave Surface, a quartic surface describing the propagation of light in an optically biaxial crystal, <math>a=1, b=0.5, c=1.5, w=1</math>.
==References==

*{{Citation | last1=Bateman | first1=H. | title= Kummer's quartic surface as a wave surface. | doi=10.1112/plms/s2-8.1.375 | year=1910 | journal=[Proceedings of the London Mathematical Society](/source/Proceedings_of_the_London_Mathematical_Society) | issn=0024-6115 | volume=8 | issue=1 | pages=375–382| url=https://zenodo.org/record/1447806 }}
*{{Citation | last1=Cayley | first1=Arthur | author1-link=Arthur Cayley | title=Sur la surface des ondes | id=Collected papers vol 1 pages 302–305 | year=1846 | journal=Journal de Mathématiques Pures et Appliquées | volume=11 | pages=291–296}}
* [Fresnel, A.](/source/Augustin-Jean_Fresnel) (1822), "Second supplément au mémoire sur la double réfraction" (signed 31&nbsp;March 1822, submitted 1&nbsp;April 1822),{{tsp}} in{{tsp}} [H.&nbsp;de&nbsp;Sénarmont](/source/Henri_Hureau_de_S%C3%A9narmont), [É.&nbsp;Verdet](/source/%C3%89mile_Verdet), and L.&nbsp;Fresnel (eds.), ''Oeuvres complètes d'Augustin Fresnel'', Paris: Imprimerie Impériale (3&nbsp;vols., 1866–70), [https://books.google.com/books?id=g6tzUG7JmoQC vol.{{nnbsp}}2&nbsp;(1868)], pp.{{nnbsp}}369–442, especially pp.&nbsp;369 (date ''présenté''), 386–8 (eq.{{nnbsp}}4), 442 (signature and date).
*{{Citation | last1=Knörrer | first1=H. |author-link=Horst Knörrer | title=Arithmetik und Geometrie | publisher=Birkhäuser | location=Basel, Boston, Berlin | series=Math. Miniaturen | isbn=978-3-7643-1759-1 | mr=879281 | year=1986 | volume=3 | chapter=Die Fresnelsche Wellenfläche | pages=[https://archive.org/details/arithmetikundgeo0000unse/page/115 115–141] | chapter-url=https://archive.org/details/arithmetikundgeo0000unse/page/115 }}
*{{Citation | last1=Love | first1=A. E. H. | title=A treatise on the Mathematical Theory of Elasticity | orig-year=1927 | url=https://books.google.com/books?id=ViebCriF-ssC | publisher=Dover Publications, New York | isbn=978-0-486-60174-8 |mr=0010851 | year=2011}}

==External links==
*[http://enriques.mathematik.uni-mainz.de/csh/galeries/complexSurfaces/fresnel.html Fresnel wave surface]

Category:Algebraic surfaces
Category:Complex surfaces
Category:Waves
Category:1822 in science

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Adapted from the Wikipedia article [Wave surface](https://en.wikipedia.org/wiki/Wave_surface) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Wave_surface?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.