Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 2 / 20 / a / Solution

Yes. The wave equation is linear and homogeneous. For $\mathcal{W}={\partial }_x^2-c^{-2}{\partial }_t^2$ and constants $A,B$,If both terms on the right vanish, their combination is also a solution. Thus the superposition principle applies, and it also preserves homogeneous fixed-end boundary conditions.