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For a regular self-adjoint Sturm-Liouville problem, choose its eigenfunctions $y_n$ orthogonal in the weighted Hilbert space $L^2((a,b),wdx)$. For real or complex $f$ in that space,The expansion converges in the weighted L2 norm. Pointwise, uniform or differentiated convergence needs extra assumptions; it does not follow from square integrability alone. The conjugation fixes the complex inner-product convention.

For the Dirichlet boundary conditions on $(0,\pi )$ with $p=w=1$, $q=0$, this is the Fourier sine series: $y_n=\sin (nx)$ and $c_n=(2/\pi ){\int }_0^{\pi }f(x)\sin (nx)dx$. The ordinary sine basis is therefore one instance of a general spectral expansion rather than an unrelated Fourier trick.

For $y_{tt}+2k{y}_t=c^2{y}_{xx}$ with fixed endpoints, initial displacement zero and initial velocity a Dirac delta at $a\in (0,l)$, the Fourier sine series coefficient of that velocity is $2\sin (n\pi a/l)/l$. Each coefficient evolves by $T_n^{\prime \prime }+2k{T}_n^{\prime }+(n\pi c/l{)}^2{T}_n=0$. Here ${\omega }_n^2=(n\pi c/l{)}^2-k^2$. At critical damping, replace $\sin ({\omega }_{n}t)/{\omega }_n$ by $t$; for an overdamped mode replace it by $\sinh (\sqrt{k^2-(n\pi c/l{)}^2}t)/\sqrt{k^2-(n\pi c/l{)}^2}$. The initial conditions hold in the sense of distributions.