---
tag: 90F3NWRHVD
kind: concept
name: Geometric optics
msc: [78A05, 35L05, 35B40]
sources:
  - url: https://en.wikipedia.org/wiki/Geometrical_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|HGMX1QGTAZ]], whose characteristics are the light rays.
The amplitudes $a_k$ solve [[transport equations|PVD7VZJZ9B]] 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.

## Explanations

### L1 human

Light is a wave, yet in daily life it seems to travel in straight lines called rays.
This is because its wavelength is tiny compared with the objects it meets.
Geometric optics makes this precise: for very short waves, the crests of the wave move along rays, and the strength of the wave is carried along each ray.
In a medium where the speed is constant, the rays are straight lines; where the speed varies, as in a mirage, the rays bend.