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  "body": "\n## Statement\n\nLet $c \u003e 0$, let $g \\in C^2(\\R)$ and let $h \\in C^1(\\R)$.\nDefine $u \\colon \\R \\times [0, \\infty) \\to \\R$ by\n$$u(x, t) = \\frac{g(x + ct) + g(x - ct)}{2} + \\frac{1}{2c} \\int_{x - ct}^{x + ct} h(s)\\, ds.$$\nThen $u \\in C^2(\\R \\times [0, \\infty))$, $u$ solves the [[wave equation|5EHD2XBQDG]] $u_{tt} = c^2 u_{xx}$ on $\\R \\times (0, \\infty)$, and $u(x, 0) = g(x)$, $u_t(x, 0) = h(x)$ for all $x \\in \\R$.\nMoreover, $u$ is the only $C^2$ solution of this initial value problem.\n\n## Remarks\n\nThe change of variables $\\xi = x + ct$, $\\eta = x - ct$ turns the equation into $u_{\\xi\\eta} = 0$.\nHence every $C^2$ solution has the form $u = F(x + ct) + G(x - ct)$, and the two initial conditions determine $F$ and $G$ up to a constant that cancels.\nThis argument also proves uniqueness.\nExample: with $g = 0$ and $h = \\cos$, the formula gives $u(x, t) = \\frac{1}{2c}\\big(\\sin(x + ct) - \\sin(x - ct)\\big) = \\frac{1}{c}\\cos x \\sin(ct)$, a standing wave.\nThe value $u(x, t)$ depends only on the data in the interval $[x - ct, x + ct]$, which shows finite speed of propagation in one dimension.\nIf $g$ is only continuous, the formula still defines a function, but it is a weak solution rather than a classical one.\n\n## Explanations\n\n### L1 human\n\nThe formula solves the vibrating string problem on an infinitely long string in closed form.\nThe initial shape $g$ splits into two halves, one moving left and one moving right at speed $c$.\nThe initial velocity $h$ adds the average of $h$ over the interval that a signal can reach in time $t$.\nSo what happens at a point depends only on what happened nearby, within distance $ct$.\n\n### L2 human\n\nd'Alembert's formula is the first explicit solution formula in a PDE course and the basis for the other dimensions.\nBy the method of spherical means, it gives Kirchhoff's formula in dimension $3$, and then by descent Poisson's formula in dimension $2$.\nOn the half line, the formula combined with an odd or even reflection of the data solves the problems with Dirichlet or Neumann boundary condition.\nFor the inhomogeneous equation $u_{tt} - c^2 u_{xx} = f$, Duhamel's principle adds the term $\\frac{1}{2c}\\int_0^t \\int_{x - c(t - s)}^{x + c(t - s)} f(y, s)\\, dy\\, ds$.\n",
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