= Solution
An <even function> satisfying the spatial equation has $X=C\cos(kx)$. The <Dirichlet boundary conditions> require $\cos(kL)=0$, so the allowed <wavenumbers> are
$$
k_n=\frac{n\pi}{2L},\qquad n=1,3,5,\ldots.
$$
There is no nonzero zero-wavenumber mode. With $T=e^{-\alpha t/2}S$, the time equation becomes $S''+\omega_n^2S=0$, where $\omega_n^2=c^2k_n^2-\alpha^2/4$. Thus <superposition> of the even <normal modes> gives
$$
\boxed{u(x,t)=\sum_{\substack{n\geq1\\ n\ {
m odd}}}e^{-\alpha t/2}\cos(k_nx)\operatorname{Re}\left(A_ne^{i\omega_nt}+B_ne^{-i\omega_nt}\right).}
$$
This uses completeness of the even <Fourier series> modes. Imaginary $\omega_n$ describe <overdamping>. At <critical damping>, $\omega_n=0$, the two displayed exponentials coincide: the complete time factor must instead be $e^{-\alpha t/2}(C_n+D_nt)$. This repeated-root exception is necessary for a genuinely general solution.
Back to article page
