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  "body": "\n## Summary\n\nIn PDE theory, geometric optics studies high-frequency solutions of wave equations.\nIt uses the ansatz $u = e^{i\\varphi/\\varepsilon}\\big(a_0 + \\varepsilon a_1 + \\cdots\\big)$ with a small wavelength $\\varepsilon$.\nThe phase $\\varphi$ solves a first-order nonlinear equation, the [[eikonal equation|HGMX1QGTAZ]], whose characteristics are the light rays.\nThe amplitudes $a_k$ solve [[transport equations|PVD7VZJZ9B]] along these rays.\nIt explains how the ray picture of light emerges from the wave picture, and it underlies the WKB method and microlocal analysis.\n\n## Explanations\n\n### L1 human\n\nLight is a wave, yet in daily life it seems to travel in straight lines called rays.\nThis is because its wavelength is tiny compared with the objects it meets.\nGeometric 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.\nIn a medium where the speed is constant, the rays are straight lines; where the speed varies, as in a mirage, the rays bend.\n",
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