11.5.1.1 d'Alembert method

Problem: Solve the problem of oscillations of an infinite string using the d'Alembert method. \[ \begin{array}{l} \frac{\partial^{2} u}{\partial t^{2}}=4 \frac{\partial^{2} u}{\partial x^{2}}, \\ \left.u(x, t)\right|_{t=0}=\sin \frac{x}{2},\left.\frac{\partial u}{\partial t}\right|_{t=0}=\cos \frac{x}{2},-\infty