# Wave surface

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In mathematics, Fresnel's wave surface, found by Augustin-Jean Fresnel in 1822, is a quartic surface describing the propagation of light in an optically biaxial crystal. Wave surfaces are special cases of tetrahedroids which are in turn special cases of Kummer surfaces.

Augustin-Jean Fresnel was a French civil engineer and physicist whose research in optics led to the almost unanimous acceptance of the wave theory of light, excluding any remnant of Newton's corpuscular theory, from the late 1830s  until the end of the 19th century. He is perhaps better known for inventing the catadioptric (reflective/refractive) Fresnel lens and for pioneering the use of "stepped" lenses to extend the visibility of lighthouses, saving countless lives at sea. The simpler dioptric stepped lens, first proposed by Count Buffon  and independently reinvented by Fresnel, is used in screen magnifiers and in condenser lenses for overhead projectors.

In mathematics, especially in algebraic geometry, a quartic surface is a surface defined by an equation of degree 4.

In algebraic geometry, a tetrahedroid is a special kind of Kummer surface studied by Cayley (1846), with the property that the intersections with the faces of a fixed tetrahedron are given by two conics intersecting in four nodes. Tetrahedroids generalize Fresnel's wave surface.

In projective coordinates (w:x:y:z) the wave surface is given by

${\displaystyle {\frac {a^{2}x^{2}}{x^{2}+y^{2}+z^{2}-a^{2}w^{2}}}+{\frac {b^{2}y^{2}}{x^{2}+y^{2}+z^{2}-b^{2}w^{2}}}+{\frac {c^{2}z^{2}}{x^{2}+y^{2}+z^{2}-c^{2}w^{2}}}=0}$

## Related Research Articles

Diffraction refers to various phenomena that occur when a wave encounters an obstacle or a slit. It is defined as the bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a secondary source of the propagating wave. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1660.

The Fresnel equations describe the reflection and transmission of light when incident on an interface between different optical media. They were deduced by Augustin-Jean Fresnel who was the first to understand that light is a transverse wave, even though no one realized that the "vibrations" of the wave were electric and magnetic fields. For the first time, polarization could be understood quantitatively, as Fresnel's equations correctly predicted the differing behaviour of waves of the s and p polarizations incident upon a material interface.

Total Internal Reflection (TIR) is the phenomenon that makes the water-to-air surface in a fish-tank look like a perfectly silvered mirror when viewed from below the water level (Fig. 1). Technically, TIR is the total reflection of a wave incident at a sufficiently oblique angle on the interface between two media, of which the second ("external") medium is transparent to such waves but has a higher wave velocity than the first ("internal") medium. TIR occurs not only with electromagnetic waves such as light waves and microwaves, but also with other types of waves, including sound and water waves. In the case of a narrow train of waves, such as a laser beam, we tend to speak of the total internal reflection of a "ray" (Fig. 2).

Fermat's principle, also known as the principle of least time, is the link between ray optics and wave optics. In its original "strong" form, Fermat's principle states that the path taken by a ray between two given points is the path that can be traversed in the least time. In order to be true in all cases, this statement must be weakened by replacing the "least" time with a time that is "stationary" with respect to variations of the path — so that a deviation in the path causes, at most, a second-order change in the traversal time. To put it loosely, a ray path is surrounded by close paths that can be traversed in very close times. It can be shown that this technical definition corresponds to more intuitive notions of a ray, such as a line of sight or the path of a narrow beam.

Baron Siméon Denis Poisson FRS FRSE was a French mathematician, engineer, and physicist who made many scientific advances.

In algebra, a quartic function is a function of the form

In algebraic geometry, a Kummer quartic surface, first studied by Kummer (1864), is an irreducible nodal surface of degree 4 in with the maximal possible number of 16 double points. Any such surface is the Kummer variety of the Jacobian variety of a smooth hyperelliptic curve of genus 2; i.e. a quotient of the Jacobian by the Kummer involution x ↦ −x. The Kummer involution has 16 fixed points: the 16 2-torsion point of the Jacobian, and they are the 16 singular points of the quartic surface. Resolving the 16 double points of the quotient of a torus by the Kummer involution gives a K3 surface with 16 disjoint rational curves; these K3 surfaces are also sometimes called Kummer surfaces.

In mathematics, a Dupin cyclide or cyclide of Dupin is any geometric inversion of a standard torus, cylinder or double cone. In particular, these latter are themselves examples of Dupin cyclides. They were discovered by Charles Dupin in his 1803 dissertation under Gaspard Monge. The key property of a Dupin cyclide is that it is a channel surface in two different ways. This property means that Dupin cyclides are natural objects in Lie sphere geometry.

In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity. There are several common standard forms of confluent hypergeometric functions:

In optics, the Fresnel diffraction equation for near-field diffraction is an approximation of the Kirchhoff–Fresnel diffraction that can be applied to the propagation of waves in the near field. It is used to calculate the diffraction pattern created by waves passing through an aperture or around an object, when viewed from relatively close to the object. In contrast the diffraction pattern in the far field region is given by the Fraunhofer diffraction equation.

In fluid dynamics, a Stokes wave is a non-linear and periodic surface wave on an inviscid fluid layer of constant mean depth. This type of modelling has its origins in the mid 19th century when Sir George Stokes – using a perturbation series approach, now known as the Stokes expansion – obtained approximate solutions for non-linear wave motion.

In fluid dynamics, Airy wave theory gives a linearised description of the propagation of gravity waves on the surface of a homogeneous fluid layer. The theory assumes that the fluid layer has a uniform mean depth, and that the fluid flow is inviscid, incompressible and irrotational. This theory was first published, in correct form, by George Biddell Airy in the 19th century.

In fluid dynamics, the mild-slope equation describes the combined effects of diffraction and refraction for water waves propagating over bathymetry and due to lateral boundaries—like breakwaters and coastlines. It is an approximate model, deriving its name from being originally developed for wave propagation over mild slopes of the sea floor. The mild-slope equation is often used in coastal engineering to compute the wave-field changes near harbours and coasts.

In projective geometry, a desmic system is a set of three tetrahedra in 3-dimensional projective space, such that any two are desmic,. It was introduced by Stephanos (1879). The three tetrahedra of a desmic system are contained in a pencil of quartic surfaces. The name "desmic" comes from the Greek word δεσμός, meaning band or chain, referring to the pencil of quartics.

In fluid dynamics, the radiation stress is the depth-integrated – and thereafter phase-averaged – excess momentum flux caused by the presence of the surface gravity waves, which is exerted on the mean flow. The radiation stresses behave as a second-order tensor.

Spherical wave transformations leave the form of spherical waves as well as the laws of optics and electrodynamics invariant in all inertial frames. They were defined between 1908 and 1909 by Harry Bateman and Ebenezer Cunningham, with Bateman giving the transformation its name. They correspond to the conformal group of "transformations by reciprocal radii" in relation to the framework of Lie sphere geometry, which were already known in the 19th century. Time is used as fourth dimension as in Minkowski space, so spherical wave transformations are connected to the Lorentz transformation of special relativity, and it turns out that the spacetime conformal group includes the Lorentz group and the Poincaré group as subgroups. However, only the Lorentz/Poincaré groups represent symmetries of all laws of nature including mechanics, whereas the conformal group is related to certain areas such as electrodynamics. In addition, it can be shown that the conformal group of the plane is isomorphic to the Lorentz group.

In fluid dynamics, a trochoidal wave or Gerstner wave is an exact solution of the Euler equations for periodic surface gravity waves. It describes a progressive wave of permanent form on the surface of an incompressible fluid of infinite depth. The free surface of this wave solution is an inverted (upside-down) trochoid – with sharper crests and flat troughs. This wave solution was discovered by Gerstner in 1802, and rediscovered independently by Rankine in 1863.

## References

• Bateman, H. (1910), "Kummer's quartic surface as a wave surface.", Proceedings of the London Mathematical Society , 8 (1): 375–382, doi:10.1112/plms/s2-8.1.375, ISSN   0024-6115
• Cayley, Arthur (1846), "Sur la surface des ondes", Journal de Mathématiques Pures et Appliquées, 11: 291–296, Collected papers vol 1 pages 302–305
• Fresnel, A. (1822), "Second supplément au mémoire sur la double réfraction" (signed 31 March 1822, submitted 1 April 1822), in H. de Sénarmont, É. Verdet, and L. Fresnel (eds.), Oeuvres complètes d'Augustin Fresnel, Paris: Imprimerie Impériale (3 vols., 1866–70), vol.2 (1868), pp.369–442, especially pp. 369 (date présenté), 386–8 (eq.4), 442 (signature and date).
• Knörrer, H. (1986), "Die Fresnelsche Wellenfläche", Arithmetik und Geometrie, Math. Miniaturen, 3, Basel, Boston, Berlin: Birkhäuser, pp. 115–141, ISBN   978-3-7643-1759-1, MR   0879281
• Love, A. E. H. (2011) [1927], A treatise on the Mathematical Theory of Elasticity, Dover Publications, New York, ISBN   978-0-486-60174-8, MR   0010851

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Arthur Cayley was a prolific British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics.