= Solution
The <wave equation> is a homogeneous <linear partial differential equation>. Consequently, if $\psi_1$ and $\psi_2$ solve it, every constant <linear combination> $A\psi_1+B\psi_2$ also solves it. This is the <superposition principle>, because <partial differentiation> distributes over the <linear combination>.
The same statement holds for the homogeneous <Neumann boundary conditions>. <Initial conditions> combine in the same way; an arbitrary combination need not retain a fixed nonzero initial displacement.
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