Consider the 1D wave equation:

Like most of these problems, I like to visualize what's going on and arrive at an intuitive interpretation of an equation, in this case, the wave equation:

Write down the straightforward finite-difference approximation.
I've made more than one attempt at writing down the straightforward finite-difference approximation (right image)
for the 1D wave equation.The answer made more sense to me after the second try (seen on the left image):

What order approximation is this in time and in space?
Second order x and t.

Use the von Neumann stability crtierion to find the mode amplitudes:

Numerically solve the wave equation. Using Mathematica: