= Solution
Apply <separation of variables> to each mode of the <damped wave equation> $u_{tt}+2k^2u_t=c^2u_{xx}$, assuming $L,c>0$. The spatial <ordinary differential equation> is $X_n''+(\lambda_n/c)^2X_n=0$. Its general solution is $C\sin(\lambda_n x/c)+D\cos(\lambda_n x/c)$. The boundary condition at $0$ gives $D=0$; a nonzero mode at $L$ requires
$$
\boxed{\lambda_n=\frac{n\pi c}L,\quad n=1,2,\ldots,\qquad X_n(x)=\sin\frac{n\pi x}L.}
$$
Only odd $n=2m+1$ occur in the initial <Fourier sine series>. Put $\omega_m=(2m+1)\pi c/L$ and $d_m=\sqrt{k^4-\omega_m^2}$. The temporal equation $T_m''+2k^2T_m'+\omega_m^2T_m=0$ has roots $-k^2\pm d_m$. Define
$$
\boxed{\alpha_m=k^2-d_m,\qquad\beta_m=k^2+d_m.}
$$
For distinct roots, let the full amplitude be $q_m(t)=A_me^{-\alpha_mt}+B_me^{-\beta_mt}$. The initial displacement and zero initial velocity impose $A_m+B_m=F_m$ and $\alpha_mA_m+\beta_mB_m=0$. Solving gives
$$
\boxed{A_m=\frac{\beta_mF_m}{\beta_m-\alpha_m},\qquad
B_m=-\frac{\alpha_mF_m}{\beta_m-\alpha_m}.}
$$
Thus the physically consistent superposition is
$$
\boxed{u(x,t)=\sum_{m=0}^\infty q_m(t)\sin\frac{(2m+1)\pi x}L.}
$$
For $\omega_m<k^2$ there is <overdamping>, with two real decaying exponentials. For $\omega_m>k^2$ there is <underdamping>; set $\nu_m=\sqrt{\omega_m^2-k^4}$; the equivalent real expression is
$$
q_m(t)=F_me^{-k^2t}\left(\cos(\nu_mt)+\frac{k^2}{\nu_m}\sin(\nu_mt)\right).
$$
At <critical damping>, the repeated-root case $\omega_m=k^2$, take the limit, giving $q_m(t)=F_m(1+k^2t)e^{-k^2t}$. These formulas all satisfy $q_m(0)=F_m$ and $q_m'(0)=0$.
Three source qualifications matter. First, the sum must start at $m=0$, for the same reason as in (b). Second, the printed $X(x)=u_0(x)$ cannot hold for a single separated mode: the tent has $X''=0$ on each open linear segment but $X\ne0$, contradicting $X''=-(\lambda_n/c)^2X$ for positive $\lambda_n$, and its peak is not twice differentiable. Separation applies mode by mode, with the initial tent recovered by superposition. Third, no finite “large” $k$ makes every mode overdamped, because $\omega_m\to\infty$. The two-exponential expression permits complex conjugate roots, while a repeated root requires its displayed limiting form. The tent initial data are naturally interpreted through the convergent modal solution, rather than as a single smooth separated product.
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