Wave equation

In the previous tutorial, we learned about Fourier transforms and how they can be used to identify what frequencies are present in a function and how they are especially useful for picking out the frequency components of sums of sine waves.

We will now study the partial differential equation that models wave propagation, and that is the aptly named wave equation.

The wave equation is a PDE that has the general form Laplacian of u equals one over c squared times the second partial derivative of u with respect to time.

Recall that the Laplacian operator can be expressed as a sum of second partial derivatives with respect to position, which in 3D Cartesian coordinates are x, y, and z.

So in the most general case, the wave equation is a PDE with four independent variables: x, y, z, and time t.

We will initially focus on the 1D case of the wave equation, and that will be uxx equals one over c squared times utt, where c is a constant representing the speed of the waves, and we can verify that the wave equation is a hyperbolic PDE, unlike the elliptic Laplace equation we looked at previously.

We will show two ways to solve the wave equation, first using separation of variables and second using Fourier transforms, both of which we covered in the previous tutorials.

For separation of variables, we assume that our solution u of x, t is the product capital X of x times capital T of t.