11.5.2.22 Fourier method

Problem: Find the solution of the Cauchy problem for the string oscillation equation: \[ \begin{array}{l} V_{t t}^{\prime \prime}=9 V_{x x}^{\prime \prime}, \quad V(x, 0)=\left\{\begin{array}{l} \frac{3 x}{40}, \quad 0 \leq x \leq 4 \\ \frac{3(8-x)}{40}, \quad 4 \leq x \leq 8 \end{array},\right. \\ V(0, t)=V(8, t)=0, \quad t \in(0,+\infty) . \end{array} \]