---
tag: 92C3S03AQZ
kind: concept
name: Hyperbolic partial differential equation
msc: [35L10, 35L25]
specializes: [GC5BDZ6QBC]
sources:
  - url: https://en.wikipedia.org/wiki/Hyperbolic_partial_differential_equation
same_as:
  wikidata: Q2627459
  wikipedia: Hyperbolic partial differential equation
---

## Summary

A hyperbolic partial differential equation is an evolution equation whose Cauchy problem is well posed for data on suitable hypersurfaces.
Disturbances travel at finite speed.
The model case is the [[wave equation|5EHD2XBQDG]].
Hyperbolic equations describe waves, vibrations and the transport of conserved quantities, and singularities of the data propagate along characteristics instead of being smoothed out.
See the [[definition|XQG2VCBMKQ]].

## Explanations

### L1 human

Pluck a guitar string: the disturbance moves along the string at a fixed speed and keeps its shape.
The equation of the vibrating string, $\partial_t^2 u = c^2 \partial_x^2 u$, is the basic hyperbolic equation.
Its solutions are sums $F(x - ct) + G(x + ct)$ of a wave moving right and a wave moving left.
A kink in the initial shape is carried along, not smoothed, which is the opposite of what the heat equation does.