Hyperbolic Equations: Definition & Explanation

[verdigris]

(d2^u/dt^2) - (delta u) = 0 is called a hyperbolic equation.

Why is this? What makes an equation a hyperbolic equation?

 

[HallsofIvy]

The partial differential equation
$$\frac{\partial^2u}{\partial t^2}= \frac{\partial^2 u}{\partial x^2}$$
and its extension to 2 or 3 space dimensions, is called "hyperbolic" in an obvious analogy with the hyperbolic equation
$$x^2- y^2= 1$$
In addition, just as the hyperbola has two asymptotes, so the hyperbolic differential equation has two "characteristic" lines that can be used to solve the equation.

The heat (or diffusion) equation,
$$\frac{\partial u}{\partial t}= \frac{\partial u^2}{\partial x^2}$$
has only a single characteristic line and is a "parabolic" equation.

Laplace's equation,
$$\frac{\partial^2 u}{\partial x^2}+ \frac{\partial^2 u}{\partial y^2}= 0$$
has no characteristic lines and is an "elliptic" equation.