---
tag: S977PT8S3S
kind: definition
name: Wave equation
msc: [35L05, 35L15]
defines: 5EHD2XBQDG
uses: [92C3S03AQZ, 0E0B6RA8K0, 09EGC6F8PT]
sources:
  - url: https://en.wikipedia.org/wiki/Wave_equation
---

## Definition

Let $n \ge 1$, let $c > 0$, and let $\Omega \subseteq \R^n$ be open.
Let $I \subseteq \R$ be an interval.
The *wave equation* with speed $c$ is the [[hyperbolic partial differential equation|92C3S03AQZ]] for $u \colon \Omega \times I \to \R$:
$$\frac{\partial^2 u}{\partial t^2} = c^2 \Delta u,$$
where $\Delta = \sum_{i=1}^{n} \partial^2/\partial x_i^2$ is the [[Laplacian|0E0B6RA8K0]] in the space variables.
The *inhomogeneous wave equation* is $u_{tt} - c^2 \Delta u = f$ for a given function $f$.

## Remarks

Example: for $n = 1$ and any $F \in C^2(\R)$, the function $u(x, t) = F(x - ct)$ is a solution, since $u_{tt} = c^2 F''(x - ct)$ and $u_{xx} = F''(x - ct)$.
Non-example: $u(x, t) = x^2$ fails for $n = 1$, since $u_{tt} = 0$ but $c^2 u_{xx} = 2c^2 \ne 0$.
The operator $\partial_t^2 - c^2\Delta$ is called the d'Alembertian or wave operator.
The Cauchy problem prescribes $u(\cdot, 0) = g$ and $u_t(\cdot, 0) = h$.