Is the wave equation a hyperbolic, parabolic, or elliptic PDE?

Question

I am a very beginner of PDEs. I want to study the wave equation in 1D and 2D for numerical methods. One basic question is which type the wave equation is. Elliptic, parabolic, or hyperbolic?

Answer 1

Generally speaking, wave equations are hyperbolic. They have the similar form that $$\frac{\partial^2u}{\partial t^2}=a^2\Delta u,$$ where $\Delta$ is the Laplacian and $u$ is the displacement of the wave.

Typical examples are acoustic wave, elastic wave, and electromagnetic.

In one dimensional, the equation is written as $$\frac{\partial^2u}{\partial t^2}=a^2\frac{\partial^2u}{\partial x^2}.$$

The general solution is $$u(x,t)=u(x-at)$$ or $$u(x,t)=u(x+at).$$