= Solution
Take the spatial interval $0<x<L$ with $L>0$, and write $v=|c|>0$. A separated nonzero solution satisfies
$$
\frac{X''}{X}=\frac{T''}{v^2T}=-\lambda.
$$
The zero-endpoint <Dirichlet boundary conditions> rule out nonzero solutions for $\lambda\leq0$: linear or hyperbolic spatial factors cannot vanish at both ends. For $\lambda>0$, they give $X_n=\sin(n\pi x/L)$ and $\lambda_n=(n\pi/L)^2$, $n\geq1$. The temporal factors solve $T_n''+\omega_n^2T_n=0$, with the <fixed-end standing wave frequencies> $\omega_n=n\pi v/L$. Superposition gives
$$
\boxed{\psi(x,t)=\sum_{n=1}^{\infty}\left[A_n\cos\left(\frac{n\pi vt}{L}\right)+B_n\sin\left(\frac{n\pi vt}{L}\right)\right]\sin\left(\frac{n\pi x}{L}\right).}
$$
The constants are arbitrary subject to the convergence and regularity required of the solution. For initial displacement $f$ and initial <velocity> $g$, their <Fourier sine series> give $A_n=(2/L)\int_0^Lf(x)\sin(n\pi x/L)dx$ and $B_n=(2/(L\omega_n))\int_0^Lg(x)\sin(n\pi x/L)dx$. Smooth compatible data give classical solutions; weaker data have the corresponding energy interpretation. A negative printed $L$ describes the interval with endpoints reversed and gives the same modes after using its positive length.
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