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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $u \\in C^2(\\R^n \\times [0, \\infty))$ solve the [[wave equation|5EHD2XBQDG]] $u_{tt} = \\Delta u$ on $\\R^n \\times (0, \\infty)$.\nLet $x_0 \\in \\R^n$ and $t_0 \u003e 0$.\nSuppose that $u(x, 0) = u_t(x, 0) = 0$ for all $x$ with $\\abs{x - x_0} \\le t_0$.\nThen $u(x, t) = 0$ for all $(x, t)$ in the cone $\\set{(x, t) : 0 \\le t \\le t_0, \\ \\abs{x - x_0} \\le t_0 - t}$.\n\n## Remarks\n\nThe proof uses the [[energy method|G8J0YQRPA3]] on shrinking balls.\nLet $e(t) = \\frac{1}{2}\\int_{B(x_0, t_0 - t)} \\big(u_t^2 + \\abs{\\nabla u}^2\\big)\\, dx$ for $0 \\le t \\le t_0$.\nDifferentiating, integrating by parts and using $\\abs{u_t\\, \\partial_\\nu u} \\le \\frac{1}{2}(u_t^2 + \\abs{\\nabla u}^2)$ on the boundary sphere gives $e'(t) \\le 0$.\nSince $e(0) = 0$, it follows that $e \\equiv 0$, so $u_t$ and $\\nabla u$ vanish in the cone and $u$ equals its initial value $0$ there.\nFor the equation $u_{tt} = c^2 \\Delta u$, the same holds with the cone $\\abs{x - x_0} \\le c(t_0 - t)$.\nExample: in one dimension, d'Alembert's formula shows directly that $u(x, t)$ depends only on the data on $[x - t, x + t]$.\nBy linearity, two solutions whose data agree on $B(x_0, t_0)$ agree in the cone, so a change of the data travels with speed at most $1$.\nThe [[heat equation|VP2HN86VAG]] has no such property: a nonnegative nonzero initial temperature with compact support gives a positive temperature everywhere at every positive time.\n\n## Explanations\n\n### L1 human\n\nWaves need time to travel.\nIf nothing is happening in a ball of radius $t_0$ at the start, then the centre stays undisturbed until time $t_0$, because no disturbance from outside can arrive sooner.\nThe undisturbed region is a cone in space-time that shrinks at the wave speed.\nThis is the mathematical form of the fact that sound or light has a finite speed.\n\n### L2 human\n\nFinite speed of propagation is the characteristic feature of [[hyperbolic equations|92C3S03AQZ]], in contrast with parabolic and elliptic ones.\nIt gives the domain of dependence of a point, the backward cone, and the domain of influence of a region of data, the forward cone.\nThe energy proof works for general second-order hyperbolic equations with variable coefficients, with the cone replaced by a region bounded by characteristic surfaces.\nIt also proves uniqueness for the Cauchy problem and allows one to solve the problem locally and patch the local solutions together.\n",
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