Geometric optics

Summary

In PDE theory, geometric optics studies high-frequency solutions of wave equations. It uses the ansatz $u = e^{i\varphi/\varepsilon}\big(a_0 + \varepsilon a_1 + \cdots\big)$ with a small wavelength $\varepsilon$. The phase $\varphi$ solves a first-order nonlinear equation, the eikonal equation, whose characteristics are the light rays. The amplitudes $a_k$ solve transport equations along these rays. It explains how the ray picture of light emerges from the wave picture, and it underlies the WKB method and microlocal analysis.