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

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

$$