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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\R^n$ be open and let $c \\colon \\Omega \\to (0, \\infty)$ be smooth.\nConsider the reduced [[wave equation|5EHD2XBQDG]] $\\Delta u + \\frac{\\omega^2}{c(x)^2} u = 0$ at a high frequency $\\omega = 1/\\varepsilon$.\nThe *geometric optics ansatz* is a formal series\n$$u(x) = e^{i\\varphi(x)/\\varepsilon} \\sum_{k=0}^{\\infty} \\varepsilon^k a_k(x),$$\nwith a real phase $\\varphi$ and amplitudes $a_k$.\nInserting it into the equation and setting the coefficient of each power of $\\varepsilon$ to zero gives:\n\n1. the [[eikonal equation|HGMX1QGTAZ]] $\\abs{\\nabla \\varphi}^2 = c(x)^{-2}$, from the power $\\varepsilon^{-2}$;\n2. the [[transport equation|PVD7VZJZ9B]] $2\\nabla\\varphi \\cdot \\nabla a_0 + (\\Delta\\varphi)\\, a_0 = 0$, from the power $\\varepsilon^{-1}$;\n3. the transport equations $2\\nabla\\varphi \\cdot \\nabla a_k + (\\Delta\\varphi)\\, a_k = i\\, \\Delta a_{k-1}$ for $k \\ge 1$.\n\n*Geometric optics* is the approximation of $u$ by such a series, with $\\varphi$ and the $a_k$ solving these equations.\nThe rays are the integral curves of the [[gradient|0ABK9V9Y64]] $\\nabla\\varphi$.\n\n## Remarks\n\nExample: with $c \\equiv 1$, the phase $\\varphi(x) = e \\cdot x$ with a unit vector $e$ satisfies the eikonal equation, and $a_0 = 1$, $a_k = 0$ for $k \\ge 1$ satisfy the transport equations.\nThe series reduces to the exact solution $e^{i e \\cdot x/\\varepsilon}$, a plane wave whose rays are straight lines in the direction $e$.\nNon-example: $\\varphi(x) = 2\\, e \\cdot x$ with $c \\equiv 1$ and $\\abs{e} = 1$ fails clause 1, since $\\abs{\\nabla \\varphi}^2 = 4$.\nThe eikonal equation usually has smooth solutions only locally; where rays cross, at caustics, the approximation breaks down and other ansätze are needed.\nThe same procedure applies to time-dependent hyperbolic equations, with phases $\\varphi(x, t)$.\n",
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