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  "body": "\n## Summary\n\nA hyperbolic partial differential equation is an evolution equation whose Cauchy problem is well posed for data on suitable hypersurfaces.\nDisturbances travel at finite speed.\nThe model case is the [[wave equation|5EHD2XBQDG]].\nHyperbolic equations describe waves, vibrations and the transport of conserved quantities, and singularities of the data propagate along characteristics instead of being smoothed out.\nSee the [[definition|XQG2VCBMKQ]].\n\n## Explanations\n\n### L1 human\n\nPluck a guitar string: the disturbance moves along the string at a fixed speed and keeps its shape.\nThe equation of the vibrating string, $\\partial_t^2 u = c^2 \\partial_x^2 u$, is the basic hyperbolic equation.\nIts solutions are sums $F(x - ct) + G(x + ct)$ of a wave moving right and a wave moving left.\nA kink in the initial shape is carried along, not smoothed, which is the opposite of what the heat equation does.\n",
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