 Solution

Use separation of variables, $y(x,t)=X(x)Q(t)$. The wave equation implies $X^{\prime \prime }/X=\ddot{Q}/(c^{2}Q)=-k^2$, giving $X^{\prime \prime }+k^{2}X=0$ and $\ddot{Q}+c^2{k}^{2}Q=0$. The fixed-end boundary conditions $X(0)=X(l)=0$ selectEach such product is a standing wave. A general released string is a superposition of these normal modes, not necessarily a single separated standing wave. Absence of damping keeps their amplitudes constant.