What is "wave surface"

Wave \Wave\, n. [From Wave, v.; not the same word as OE. wawe, waghe, a wave, which is akin to E. wag to move. [root]138. See Wave, v. i.]

1. An advancing ridge or swell on the surface of a liquid, as of the sea, resulting from the oscillatory motion of the particles composing it when disturbed by any force their position of rest; an undulation.

   The wave behind impels the wave before.
   --Pope.

2. (Physics) A vibration propagated from particle to particle through a body or elastic medium, as in the transmission of sound; an assemblage of vibrating molecules in all phases of a vibration, with no phase repeated; a wave of vibration; an undulation. See Undulation.

3. Water; a body of water. [Poetic] ``Deep drank Lord Marmion of the wave.''
   --Sir W. Scott.

   Build a ship to save thee from the flood, I 'll furnish thee with fresh wave, bread, and wine.
   --Chapman.

4. Unevenness; inequality of surface.
   --Sir I. Newton.

5. A waving or undulating motion; a signal made with the hand, a flag, etc.

6. The undulating line or streak of luster on cloth watered, or calendered, or on damask steel.

7. Something resembling or likened to a water wave, as in rising unusually high, in being of unusual extent, or in progressive motion; a swelling or excitement, as of feeling or energy; a tide; flood; period of intensity, usual activity, or the like; as, a wave of enthusiasm; waves of applause. Wave front (Physics), the surface of initial displacement of the particles in a medium, as a wave of vibration advances. Wave length (Physics), the space, reckoned in the direction of propagation, occupied by a complete wave or undulation, as of light, sound, etc.; the distance from a point or phase in a wave to the nearest point at which the same phase occurs. Wave line (Shipbuilding), a line of a vessel's hull, shaped in accordance with the wave-line system. Wave-line system, Wave-line theory (Shipbuilding), a system or theory of designing the lines of a vessel, which takes into consideration the length and shape of a wave which travels at a certain speed. Wave loaf, a loaf for a wave offering. --Lev. viii. 27. Wave moth (Zo["o]l.), any one of numerous species of small geometrid moths belonging to Acidalia and allied genera; -- so called from the wavelike color markings on the wings. Wave offering, an offering made in the Jewish services by waving the object, as a loaf of bread, toward the four cardinal points. --Num. xviii. 11. Wave of vibration (Physics), a wave which consists in, or is occasioned by, the production and transmission of a vibratory state from particle to particle through a body. Wave surface.

   1. (Physics) A surface of simultaneous and equal displacement of the particles composing a wave of vibration.

   2. (Geom.) A mathematical surface of the fourth order which, upon certain hypotheses, is the locus of a wave surface of light in the interior of crystals. It is used in explaining the phenomena of double refraction. See under Refraction.

      Wave theory. (Physics) See Undulatory theory, under Undulatory.

In mathematics, Fresnel's wave surface, found by Augustin-Jean Fresnel in 1821, 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.

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

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