This lecture will do apply separation of variables and Fourier series in order to solve for the wave equation on a finite interval.

We look to solve $$\begin{gathered} u_{tt} = c^2 u_{xx} \\ u(0, t) = 0 = u(L, t) \\ u(x, 0) = u_0(x), \\ u_t(x, 0) = v_0(x), \end{gathered}$$

which models the deflection of a string of length $L$ given initial displacement and velocities.

In the video, we'll cover the separation of variables procedure, which follows Sec. 16.2 of the notes. The procedure is virtually identical to the case of the heat equation, with the exception that the equation for two components,

$$u(x, t) = X(t) T(t),$$

yields a second-order ODE for $T(t)$ instead of a first-order ODE. Consequently, the relevant solutions are sinusoidals for both $X$ and $T$. We will show that the separable solutions are given by

$$u_n(x, t) = \sin\left(\frac{n\pi x}{L}\right) \left[ A_n \cos\left(\frac{n\pi ct}{L}\right) + B_n \sin\left(\frac{n\pi ct}{L}\right)\right]$$

Once you sum this over all the positive integers, then you will obtain the general solution in terms of the Fourier series. To this solution, we must satisfy the initial displacement and velocities.

I want to help you imagine what the modes, $u_n$, look like. For simplicity, take the length $L = \pi$. Also, we may take $c = 1$.