Geometrical optics

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

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

where $k,\; omega$ satisfy a dispersion relation, and the amplitude $a(t,x)$ varies slowly. More precisely, the leading order solution takes the form:$a\_0(t,x)\; e^\{ivarphi(t,x)/varepsilon\}.$The phase $varphi(t,x)/varepsilon$ can be linearized to recover large wavenumber $k:=\; abla\_x\; varphi$, and frequency $omega\; :=\; -partial\_t\; varphi$. The amplitude $a\_0$ satisfies a transport equation. The small parameter $varepsilon$ enters the scene due to highly oscillatory initial conditions. Thus, when initial conditions oscillate much faster than the coefficients of the differential equation, solutions will be highly oscillatory, and transported along rays. Assuming coefficients in the differential equation are smooth, the rays will be too. In other words, refraction does not take place. The motivation for this technique comes from studying the typical scenario of light propagation where short wavelength light travels along rays that minimize (more or less) its travel time. Its full application requires tools from microlocal analysis.

A Simple Example

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

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

one looks for an asymptotic series solution of the form

:$u(t,x)\; sim\; a\_varepsilon(t,x)e^\{ivarphi(t,x)/varepsilon\}\; =\; sum\_\{j=0\}^infty\; i^j\; varepsilon^j\; a\_j(t,x)\; e^\{ivarphi(t,x)/varepsilon\}.$One may check that:$L(partial\_t,\; abla\_x)(e^\{ivarphi(t,x)/varepsilon\})\; a\_varepsilon(t,x)\; =\; e^\{ivarphi(t,x)/varepsilon\}\; left(\; left(frac\{i\}\{varepsilon\}\; ight)^2\; L(varphi\_t,\; abla\_xvarphi)a\_varepsilon\; +\; frac\{2i\}\{varepsilon\}\; V(partial\_t,\; abla\_x)a\_varepsilon\; +\; frac\{i\}\{varepsilon\}\; (a\_varepsilon\; L(partial\_t,\; abla\_x)varphi)\; +\; L(partial\_t,\; abla\_x)a\_varepsilon\; ight)$with:$V(partial\_t,\; abla\_x)\; :=\; frac\{partial\; varphi\}\{partial\; t\}\; frac\{partial\}\{partial\; t\}\; -\; c^2(x)sum\_j\; frac\{partial\; varphi\}\{partial\; x\_j\}\; frac\{partial\}\{partial\; x\_j\}$

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),:$0\; =\; L(varphi\_t,\; abla\_xvarphi)\; =\; (varphi\_t)^2\; -\; c(x)^2(\; abla\_x\; varphi)^2.$To order $varepsilon^\{-1\}$ we find that the leading order amplitude must satisfy a transport equation:$2V\; a\_0\; +\; (Lvarphi)a\_0\; =\; 0$

With the definition $k\; :\; =\; abla\_x\; varphi$, $omega\; :=\; -varphi\_t$, 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.

External links

* [http://www.math.lsa.umich.edu/~rauch/nlgonotes.pdf Online book, "Hyperbolic Partial Differential Equations and Geometrical Optics"]