The wave equation

In this chapter, we are going to solve numerically the wave equation. Unlike the previous equations we've been solving, this is a partial differential equation. For this new kind of equations, we are going to learn new methods, such as the method of lines.

Lets start by defining the problem for the 1 dimensional wave equation. We are trying to find a function f which depends of one spatial coordinate x and one temporal coordinate. All we know about f is that it satisfies the wave equation:

$$\frac{\partial^2f}{\partial x^2} = \frac{1}{v^2}\frac{\partial^2f}{\partial t^2}$$

and that it satisfies the following initial conditions:

$$\begin{align}f(x,t=0) &= g(x)\nonumber\\ \frac{\partial f}{\partial t}(x,t=0) &= h(x) \nonumber \end{align}$$

In the following sections, we are going to show ways in which one can approximate the value of this function in a discrete domain.