Separation of variables

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.

Substituting in and separating the variables leads to two Sturm-Liouville problems, and if we choose negative lambda squared as our separation constant, we will get trigonometric solutions for each of x and t, which is usually going to be the better choice for describing waves.

Without any boundary conditions, we can accept any value of lambda whatsoever, and the solution will be valid.

Intuitively, the string should vibrate at the fundamental frequency of the string, producing a tone, but let's see what the math can tell us.

We just need to take our general solution to the wave equation and use the initial condition u of x 0 equals this triangle-shaped function, as well as partial u partial t equals 0 at t equals 0, which is a boundary condition that states that the string starts from rest.

To do this, it will be easier to return to our separation of variables solution shown here.