11.5.2.17 Fourier method

Problem: Find the solution of the Cauchy problem for the string oscillation equation: \[ \begin{array}{ll} u_{t t}=16 u_{x x}, & u_{x}(0, t)=u_{x}(4, t)=0, \\ u(x, 0)=0, & u_{t}(x, 0)=12 \pi \cos 3 \pi x . \end{array} \]