### Vibrating string equation (without damping)

Vibrating string without damping is represented by the following differential equation: $$ \begin{cases} \begin{array}{l@{\ }l@{\ }l} \frac{{{\partial ^2}u}}{{\partial {t^2}}} = {a^2}\frac{{{\partial ^2}u}}{{\partial {x^2}}} , & \hspace{0.25in} 0 \leqslant x \leqslant L , 0 \leqslant t \leqslant \infty , & \hspace{0.25in} \text{string equation} ; \\ u\left( {0,t} \right) = u\left( {L,t} \right) = 0 , & \hspace{0.25in} 0 \leqslant t \leqslant \infty , & \hspace{0.25in} \text{boundary conditions} ; \\ u\left( {x,0} \right) = f\left( x \right) , \frac{{\partial u}}{{\partial t}}\left( {x,0} \right) = 0 , & \hspace{0.25in} 0 \leqslant x \leqslant L , & \hspace{0.25in} \text{initial conditions}; \end{array} \end{cases} \tag{1} $$ Splitting the string equation into two coupled equations We need to transform the equation: $$ \frac{{{\partial ^2}u}}{{\partial {t^2}}} = {a^2}\frac{{{\partial ^2}u}}{{\parti