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  "body": "\n## Definition\n\nLet $n \\ge 1$, let $c \u003e 0$, and let $\\Omega \\subseteq \\R^n$ be open.\nLet $I \\subseteq \\R$ be an interval.\nThe *wave equation* with speed $c$ is the [[hyperbolic partial differential equation|92C3S03AQZ]] for $u \\colon \\Omega \\times I \\to \\R$:\n$$\\frac{\\partial^2 u}{\\partial t^2} = c^2 \\Delta u,$$\nwhere $\\Delta = \\sum_{i=1}^{n} \\partial^2/\\partial x_i^2$ is the [[Laplacian|0E0B6RA8K0]] in the space variables.\nThe *inhomogeneous wave equation* is $u_{tt} - c^2 \\Delta u = f$ for a given function $f$.\n\n## Remarks\n\nExample: for $n = 1$ and any $F \\in C^2(\\R)$, the function $u(x, t) = F(x - ct)$ is a solution, since $u_{tt} = c^2 F''(x - ct)$ and $u_{xx} = F''(x - ct)$.\nNon-example: $u(x, t) = x^2$ fails for $n = 1$, since $u_{tt} = 0$ but $c^2 u_{xx} = 2c^2 \\ne 0$.\nThe operator $\\partial_t^2 - c^2\\Delta$ is called the d'Alembertian or wave operator.\nThe Cauchy problem prescribes $u(\\cdot, 0) = g$ and $u_t(\\cdot, 0) = h$.\n",
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