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  "body": "\n## Definition\n\nLet $F \\colon \\R \\to \\R$ be a $C^2$ function with $F'' \u003e 0$, and consider the scalar [[conservation law|8RA7AMWSAF]] $u_t + F(u)_x = 0$ on $\\R \\times (0, \\infty)$.\nLet $G$ be the inverse function of the strictly increasing function $F'$.\nLet $u_l \u003c u_r$ be two real numbers.\nThe *centered rarefaction wave* from $u_l$ to $u_r$ is the function\n$$u(x, t) = \\begin{cases} u_l \u0026 \\text{if } x \u003c F'(u_l)\\, t, \\\\ G(x/t) \u0026 \\text{if } F'(u_l)\\, t \\le x \\le F'(u_r)\\, t, \\\\ u_r \u0026 \\text{if } x \u003e F'(u_r)\\, t. \\end{cases}$$\n\n## Remarks\n\nThe function is continuous, since $G(F'(u_l)) = u_l$ and $G(F'(u_r)) = u_r$.\nInside the fan, $F'(u) = x/t$, so $u_t = -\\tfrac{x}{t^2} G'(x/t)$ and $u_x = \\tfrac1t G'(x/t)$.\nHence $u_t + F'(u)\\, u_x = 0$ there, and $u$ is a [[weak solution|SJZX81A783]] with the jump initial values $u_l$ for $x \u003c 0$ and $u_r$ for $x \u003e 0$.\nIt depends only on $x/t$, so it is self-similar.\n\nExample: for [[Burgers' equation|NEGHXN8RE1]], $F(u) = u^2/2$, so $F'(u) = u$ and $G$ is the identity.\nThe rarefaction wave from $0$ to $1$ is $u = 0$ for $x \u003c 0$, $u = x/t$ for $0 \\le x \\le t$, and $u = 1$ for $x \u003e t$.\nCheck: $u_t + u u_x = -x/t^2 + (x/t)(1/t) = 0$ inside the fan.\n\nNon-example: if $u_l \u003e u_r$, then $F'(u_l) t \u003e F'(u_r) t$ and the middle interval is empty.\nThe formula then gives no fan, and the admissible solution is a [[shock wave|JSDMGGTG7G]] instead.\n\n## Explanations\n\n### L1 human\n\nA rarefaction wave is a fan of straight lines that spreads a jump into a smooth ramp.\nAlong each line through the origin with slope $x/t = c$, the solution takes the value whose characteristic speed is $c$.\nSo the solution moves continuously from the left state to the right state.\n",
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