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In fluid dynamics, Airy wave theory (often referred to as linear 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 .
The light-blue area is the range of validity of cnoidal wave theory; light-yellow for Airy wave theory; and the dashed blue lines demarcate the required order of Stokes' wave theory. The light-gray shading gives the range extension by numerical approximations using fifth-order stream-function theory, for high waves (H > ¼ H breaking).
The term "Airy beam" derives from the Airy integral, developed in the 1830s by Sir George Biddell Airy to explain optical caustics such as those appearing in a rainbow. [1] The Airy waveform was first theorized in 1979 by M. V. Berry and Nándor L. Balázs. They demonstrated a nonspreading Airy wave packet solution to the Schrödinger equation. [2]
The Airy wave train is the only dispersionless wave in one dimensional free space. [20] In higher dimensions, other dispersionless waves are possible. [21] The Airy wave train in phase space. Its shape is a series of parabolas with the same axis, but oscillating according to the Airy function.
1841 – George Biddell Airy publishes the first correct formulation of Airy wave theory of water waves. [27] 1842 – Christian Doppler introduces the Doppler effect. 1842-1850 – Stokes completes the equations of motions of fluids, now referred as Navier–Stokes equations. He also extends Airy wave theory to non-linear Stokes wave theory. [28]
Note that the second- and third-order terms in the velocity potential Φ are zero. Only at fourth order do contributions deviating from first-order theory – i.e. Airy wave theory – appear. [6] Up to third order the orbital velocity field u = ∇Φ consists of a circular motion of the velocity vector at each position (x,z).
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The conditions for being in the far field and exhibiting an Airy pattern are: the incoming light illuminating the aperture is a plane wave (no phase variation across the aperture), the intensity is constant over the area of the aperture, and the distance from the aperture where the diffracted light is observed (the screen distance) is large ...