# Hyperbolic PDEs

A simple example of a hyperbolic PDE is the one-dimensional wave equation:

$$\frac{\partial^2 u}{\partial x^2} - \frac{\partial^2 u}{\partial t^2} = 0.$$
The discriminant of this equation is positive (B² - 4AC = 0² - (4)(1)(-1) = 4); so it is classified as hyperbolic.

The hyperbolic equation arises in various physical fields such as hydraulics, acoustics and elasticity. Hyperbolic PDEs have real characteristics which, in general, will be curved. It means that along these curves, the PDE becomes an ordinary differential equation (ODE) and can be solved along the characteristics.