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As a mathematical study, geometrical optics emerges as a short-wavelength limit for solutions to hyperbolic partial differential equations. For a less mathematical introduction, please see optics. In this short wavelength limit, it is possible to approximate the solution locally by## A Simple Example

Starting with the wave equation for $(t,x)\; in\; mathbb\{R\}timesmathbb\{R\}^n$## External links

- $u(t,x)\; approx\; a(t,x)e^\{i(kcdot\; x\; -\; omega\; t)\}$

where $k,\; omega$ satisfy a dispersion relation, and the amplitude $a(t,x)$ varies slowly. More precisely, the leading order solution takes the form

- $a\_0(t,x)\; e^\{ivarphi(t,x)/varepsilon\}.$

- $L(partial\_t,\; nabla\_x)\; u\; :=\; left(frac\{partial^2\}\{partial\; t^2\}\; -\; c(x)^2\; Delta\; right)u(t,x)\; =\; 0,\; ;;\; u(0,x)\; =\; u\_0(x),;;\; u\_t(0,x)\; =\; 0$

one looks for an asymptotic series solution of the form

- $u(t,x)\; sim\; a\_varepsilon(t,x)e^\{ivarphi(t,x)/varepsilon\}\; =\; sum\_\{j=0\}^infty\; i^j\; varepsilon^j\; a\_j(t,x)\; e^\{ivarphi(t,x)/varepsilon\}.$

- $L(partial\_t,nabla\_x)(e^\{ivarphi(t,x)/varepsilon\})\; a\_varepsilon(t,x)\; =\; e^\{ivarphi(t,x)/varepsilon\}$

- $V(partial\_t,nabla\_x)\; :=\; frac\{partial\; varphi\}\{partial\; t\}\; frac\{partial\}\{partial\; t\}\; -\; c^2(x)sum\_j\; frac\{partial\; varphi\}\{partial\; x\_j\}\; frac\{partial\}\{partial\; x\_j\}$

Plugging the series into this equation, and equating powers of $varepsilon$, we find that the most singular term $O(varepsilon^\{-2\})$ satisfies the eikonal equation (in this case called a dispersion relation),

- $0\; =\; L(varphi\_t,nabla\_xvarphi)\; =\; (varphi\_t)^2\; -\; c(x)^2(nabla\_x\; varphi)^2.$

- $2V\; a\_0\; +\; (Lvarphi)a\_0\; =\; 0$

With the definition $k\; :\; =\; nabla\_x\; varphi$, $omega\; :=\; -varphi\_t$, the eikonal equation is precisely the dispersion relation one would get by plugging the plane wave solution $e^\{i(kcdot\; x\; -\; omega\; t)\}$ into the wave equation. The value of this more complicated expansion is that plane waves cannot be solutions when the wavespeed $c$ is non-constant. However, one can show that the amplitude $a\_0$ and phase $varphi$ are smooth, so that on a local scale we have plane waves.

To justify this technique, one must show that the remaining terms are small in some sense. This can be done using energy estimates, and an assumption of rapidly oscillating initial conditions. It also must be shown that the series converges in some sense.

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Last updated on Saturday October 04, 2008 at 09:38:46 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Saturday October 04, 2008 at 09:38:46 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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