= Solution
The <wave equation> and its homogeneous <Dirichlet boundary conditions> are linear, so any finite <linear combination> of the separated <normal modes> is another solution. More strongly, the <Fourier sine basis> is complete in <L2 space> on $(0,L)$. Expanding the initial fields in this basis yields
$$
\boxed{\eta(x,t)=\sum_{n=1}^{\infty}
\bigl[A_n\cos(\omega_nt)+B_n\sin(\omega_nt)\bigr]
\sin\frac{n\pi x}{L}.}
$$
<Completeness of the Fourier sine basis> is the reason the expansion covers general data; <linearity> alone would not prove that every solution has this form. The convergence and differentiation are understood in a function space appropriate to the data. For finite <wave energy>, one can take $f\in H_0^1(0,L)$ and $g\in L^2(0,L)$, obtaining an energy solution; stronger smoothness and endpoint compatibility give a classical solution.
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