Eikonal equation

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A partial differential equation of the form

$$\sum_{i=1}^m\left(\frac{\partial\tau}{\partial x^i}\right)^2=\frac{1}{c^2(x^1,\dots,x^m)}.$$

Here $m$ is the dimension of the space and $c$ is a smooth function bounded away from zero. In applications $c$ is the speed of the wave, and the surfaces $\tau(x^1,\dots,x^m)=\mathrm{const}$ are the wave fronts. The rays (see Fermat principle) are the characteristics of the eikonal equation. This equation has a number of generalizations and analogues. In particular, one such generalization is

$$H\left(x^1,\dots,x^m,\frac{\partial\tau}{\partial x^1},\dots,\frac{\partial\tau}{\partial x^m}\right)=1,$$

where the function $H$ is homogeneous of degree 1 with respect to $\partial\tau/\partial x^1,\dots,\partial\tau/\partial x^m$ and satisfies some additional restrictions. Of considerable interest is the non-stationary analogue

$$-\frac{\partial\theta}{\partial t}+c(t,x^1,\dots,x^m)\sqrt{\sum_{i=1}^m\left(\frac{\partial\theta}{\partial x^i}\right)^2}=0.$$

This is a special case of the dispersion equations occurring in the theory of wave phenomena, which have the form

$$\frac{\partial\theta}{\partial t}=\omega(t,x^1,\dots,x^m,\theta_{x^1},\theta_{x^m}).$$

Here $\omega$ is a given function.

The mathematical theory of geometrical optics can be regarded as the theory of the eikonal equation. All forms of the eikonal equation are first-order partial differential equations. The solution of the eikonal equation may have singularities. Their theory is part of that of the singularities of differentiable mappings (see also Hamilton–Jacobi theory; Geometric approximation, and Ray method).


For a nice account of the theory of geometrical optics see [a3]; geometrical optics and pseudo-differential operator theory are treated in [a2].


[1] V.M. Babich, V.S. Buldyrev, "Asymptotic methods in problems of diffraction of short waves" , Moscow (1972) (In Russian) (Translation forthcoming: Springer)
[2] G.B. Whitham, "Linear and nonlinear waves" , Wiley (1974) MR0483954 Zbl 0373.76001
[a1] P.R. Garabedian, "Partial differential equations" , Wiley (1964) MR0162045 Zbl 0124.30501
[a2] M.E. Taylor, "Pseudodifferential operators" , Princeton Univ. Press (1981) MR0618463 Zbl 0453.47026
[a3] M. Kline, I.W. Kay, "Electromagnetic theory and geometrical optics" , Interscience (1965) MR0180094 Zbl 0123.23602
[a4] L.B. Felsen, N. Marcuvitz, "Radiation and scattering of waves" , Prentice-Hall (1973) pp. Sect. 1.7 MR0471626
How to Cite This Entry:
Eikonal equation. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.M. Babich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article