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9 Airy and Related FunctionsApplications

§9.16 Physical Applications

Airy functions are applied in many branches of both classical and quantum physics. The function Ai(x) first appears as an integral in two articles by G.B. Airy on the intensity of light in the neighborhood of a caustic (Airy (1838, 1849)). Details of the Airy theory are given in van de Hulst (1957) in the chapter on the optics of a raindrop. See also Berry (1966, 1969).

The frequent appearances of the Airy functions in both classical and quantum physics is associated with wave equations with turning points, for which asymptotic (WKBJ) solutions are exponential on one side and oscillatory on the other. The Airy functions constitute uniform approximations whose region of validity includes the turning point and its neighborhood. Within classical physics, they appear prominently in physical optics, electromagnetism, radiative transfer, fluid mechanics, and nonlinear wave propagation. Examples dealing with the propagation of light and with radiation of electromagnetic waves are given in Landau and Lifshitz (1962). Extensive use is made of Airy functions in investigations in the theory of electromagnetic diffraction and radiowave propagation (Fock (1965)). The use of Airy function and related uniform asymptotic techniques to calculate amplitudes of polarized rainbows can be found in Nussenzveig (1992) and Adam (2002). A quite different application is made in the study of the diffraction of sound pulses by a circular cylinder (Friedlander (1958)).

In fluid dynamics, Airy functions enter several topics. In the study of the stability of a two-dimensional viscous fluid, the flow is governed by the Orr–Sommerfeld equation (a fourth-order differential equation). Again, the quest for asymptotic approximations that are uniformly valid solutions to this equation in the neighborhoods of critical points leads (after choosing solvable equations with similar asymptotic properties) to Airy functions. Other applications appear in the study of instability of Couette flow of an inviscid fluid. These examples of transitions to turbulence are presented in detail in Drazin and Reid (1981) with the problem of hydrodynamic stability. The investigation of the transition between subsonic and supersonic of a two-dimensional gas flow leads to the Euler–Tricomi equation (Landau and Lifshitz (1987)). An application of Airy functions to the solution of this equation is given in Gramtcheff (1981).

Airy functions play a prominent role in problems defined by nonlinear wave equations. These first appeared in connection with the equation governing the evolution of long shallow water waves of permanent form, generally called solitons, and are predicted by the Korteweg–de Vries (KdV) equation (a third-order nonlinear partial differential equation). The KdV equation and solitons have applications in many branches of physics, including plasma physics lattice dynamics, and quantum mechanics. (Ablowitz and Segur (1981), Ablowitz and Clarkson (1991), and Whitham (1974).)

Reference to many of these applications as well as to the theory of elasticity and to the heat equation are given in Vallée and Soares (2010): a book devoted specifically to the Airy and Scorer functions and their applications in physics.

An example from quantum mechanics is given in Landau and Lifshitz (1965), in which the exact solution of the Schrödinger equation for the motion of a particle in a homogeneous external field is expressed in terms of Ai(x). Solutions of the Schrödinger equation involving the Airy functions are given for other potentials in Vallée and Soares (2010). This reference provides several examples of applications to problems in quantum mechanics in which Airy functions give uniform asymptotic approximations, valid in the neighborhood of a turning point. A study of the semiclassical description of quantum-mechanical scattering is given in Ford and Wheeler (1959a, b). In the case of the rainbow, the scattering amplitude is expressed in terms of Ai(x), the analysis being similar to that given originally by Airy (1838) for the corresponding problem in optics.

An application of the Scorer functions is to the problem of the uniform loading of infinite plates (Rothman (1954b, a)).