Underlying Mathematics
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As a mathematical study, geometrical optics emerges as a short-wavelength limit for solutions to hyperbolic partial differential equations. In this short-wavelength limit, it is possible to approximate the solution locally by
where satisfy a dispersion relation, and the amplitude varies slowly. More precisely, the leading order solution takes the form
The phase can be linearized to recover large wavenumber, and frequency . The amplitude satisfies a transport equation. The small parameter 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.
Read more about this topic: Geometrical Optics
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