View source for 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|singularities of differentiable mappings]] (see also [[Hamilton–Jacobi theory]]; [[Geometric approximation]], and [[Ray method]]).


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

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> V.M. Babich, V.S. Buldyrev, "Asymptotic methods in problems of diffraction of short waves" , Moscow (1972) (In Russian) (Translation forthcoming: Springer)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> G.B. Whitham, "Linear and nonlinear waves" , Wiley (1974) {{MR|0483954}} {{ZBL|0373.76001}} </TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> P.R. Garabedian, "Partial differential equations" , Wiley (1964) {{MR|0162045}} {{ZBL|0124.30501}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M.E. Taylor, "Pseudodifferential operators" , Princeton Univ. Press (1981) {{MR|0618463}} {{ZBL|0453.47026}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> M. Kline, I.W. Kay, "Electromagnetic theory and geometrical optics" , Interscience (1965) {{MR|0180094}} {{ZBL|0123.23602}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> L.B. Felsen, N. Marcuvitz, "Radiation and scattering of waves" , Prentice-Hall (1973) pp. Sect. 1.7 {{MR|0471626}} {{ZBL|}} </TD></TR></table>