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  "body": "\n## Definition\n\nLet $g \u003e 0$ be a constant, the gravitational acceleration.\nThe one-dimensional *shallow water equations* over a flat bottom are the system\n$$\\partial_t h + \\partial_x (h u) = 0, \\qquad \\partial_t (h u) + \\partial_x \\big( h u^2 + \\tfrac{1}{2} g h^2 \\big) = 0,$$\nfor unknown functions $h, u \\colon \\R \\times [0, \\infty) \\to \\R$ with $h \u003e 0$.\nHere $h$ is the depth of the fluid and $u$ its horizontal velocity.\nThe symbols $\\partial_t$ and $\\partial_x$ denote [[partial derivatives|09EGC6F8PT]].\n\n## Remarks\n\nThe system has the form $\\partial_t U + \\partial_x \\Phi(U) = 0$ for $U = (h, hu)$, so it is a system of [[hyperbolic conservation laws|XN081NCPF5]].\nThe eigenvalues of the Jacobian matrix of $\\Phi$ are $u \\pm \\sqrt{g h}$, which are real and distinct when $h \u003e 0$.\n\nExample: for constants $H \u003e 0$ and $U_0 \\in \\R$, the uniform flow $h = H$, $u = U_0$ is a solution, since all derivatives vanish.\n\nNon-example: $h(x, t) = 1 + x^2$ and $u = 0$ do not form a solution, because $\\partial_x(\\tfrac{1}{2} g h^2) = g h \\, \\partial_x h = 2 g x (1 + x^2)$ is not identically $0$.\n\nIn two dimensions, with velocity $(u, v)$, the equations are $\\partial_t h + \\nabla \\cdot (h u, h v) = 0$ together with two momentum equations of the same form.\nVariants add a varying bottom, friction or the Coriolis force.\n",
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