Airy

Airy

The term "Airy" can refer to several concepts, primarily associated with the work of the British astronomer and mathematician Sir George Biddell Airy. This article will explore the various contexts in which "Airy" is used, including the Airy function, Airy disk, and Airy stress function, among others.

Airy Function

The Airy function is a solution to the differential equation:

<math>y - xy = 0</math>

This equation is known as the Airy equation. The Airy functions, denoted as <math>Ai(x)</math> and <math>Bi(x)</math>, are used in physics to describe wave propagation and diffraction phenomena. They are particularly useful in quantum mechanics and optics.

Properties

• The Airy function <math>Ai(x)</math> is defined as:

<math>Ai(x) = \frac{1}{\pi} \int_0^\infty \cos\left(\frac{t^3}{3} + xt\right) \, dt</math>

• The Airy function <math>Bi(x)</math> is defined as:

<math>Bi(x) = \frac{1}{\pi} \int_0^\infty \left[ \exp\left(-\frac{t^3}{3} + xt\right) + \sin\left(\frac{t^3}{3} + xt\right) \right] \, dt</math>

• These functions are linearly independent solutions to the Airy equation.

Airy Disk

The Airy disk is a pattern of diffraction that occurs when light passes through a circular aperture. It is named after George Biddell Airy, who first described the phenomenon in 1835.

Description

• The central bright region of the diffraction pattern is known as the Airy disk.
• The radius of the Airy disk is given by:

<math>r = 1.22 \frac{\lambda}{D}</math>

where <math>\lambda</math> is the wavelength of light and <math>D</math> is the diameter of the aperture.

• The Airy disk is important in optics and astronomy as it determines the resolving power of optical systems.

Airy Stress Function

The Airy stress function is a scalar function used in the field of solid mechanics to solve problems in two-dimensional elasticity.

Definition

• The Airy stress function <math>\phi(x, y)</math> satisfies the biharmonic equation:

<math>\nabla^4 \phi = 0</math>

• The stress components in terms of the Airy stress function are:

<math>\sigma_{xx} = \frac{\partial^2 \phi}{\partial y^2}</math>

<math>\sigma_{yy} = \frac{\partial^2 \phi}{\partial x^2}</math>

<math>\sigma_{xy} = -\frac{\partial^2 \phi}{\partial x \partial y}</math>

See Also

• George Biddell Airy
• Diffraction
• Elasticity (physics)

References

• Airy, G. B. (1835). "On the Diffraction of an Object-glass with Circular Aperture". Transactions of the Cambridge Philosophical Society.
• Arfken, G. B., & Weber, H. J. (2005). Mathematical Methods for Physicists.