11.5.2.6 Fourier method

Problem: Solve the boundary-value problem for the homogeneous wave equation: \[ \begin{array}{l} u_{t t}=a^{2} u_{x x}, \quad a=1, \quad u(0, x)=\frac{x}{2}, \\ \frac{\partial u}{\partial t}(0, x)=u(t, 0)=u(t, l)=0 . \end{array} \]