 Solution

For each sine normal mode, the temporal characteristic equation is $s^2+2\lambda s+{\alpha }_n^2=0$, with roots $s=-\lambda \pm \sqrt{{\lambda }^2-{\alpha }_n^2}$. Since $\lambda L/\pi =M+1/2$, the first $M$ modes are overdamped and all higher modes are underdamped. No integer $n$ gives critical damping.

The zero initial displacement excludes the hyperbolic cosine and cosine terms. HencewhereTo impose the initial velocity, let $f_n=\frac{2}{L}{\int }_0^{L}f(x)\sin ({\alpha }_{n}x)dx$ be its Fourier sine series coefficients. Differentiating the expansion at $t=0$ gives $A_n{\Omega }_n=f_n$ for the overdamped modes and $A_n{\omega }_n=f_n$ for the underdamped modes. These determine all coefficients. Despite the hyperbolic sine, the first modes decay: ${\Omega }_n<\lambda$, so both temporal exponential rates are negative.