Damped boundary conditions and normal modes

A string of length $L$ is under tension $\tau$ and has linear mass density $\mu$. The string is free to move along the $y$-direction at one end $(x = L)$ while the other end $(x = 0)$ is attached to a massless ring that can slide up and down a rod but which is also attached to a shock absorber which exerts a vertical drag force on the ring $\vec F = - b \vec v$. (See Figure 1.)

Figure 1: Damped boundary conditions on a string at $x=0$.

(a)

Use $\sum \vec{F}^{(\mathrm{ext})} = m\vec{a}^{(\mathrm{c\,of\,m})}$ to find a relationship between $\partial y(x=0,t)/\partial x$ and $\partial y(x=0,t)/\partial t$.

(b)

Consider the limit $b \rightarrow 0$ (specifically, $b << \tau$ or $b/\tau \rightarrow 0$). What does your answer to (a) tell you about the motion and/or the shape of the string at $x=0$ in this limit? What type of familiar boundary condition (``free'' or ``fixed'') does this represent? In terms of the string length $L$, what are the allowed wave vectors $k$ for the normal modes in this case?

(c)

Now, consider the limit $b/\tau \rightarrow \infty$. What does your answer to (a) tell you about the motion and/or the shape of the string at $x=0$? What type of familiar boundary condition (``free'' or ``fixed'') does this represent? In terms of the string length $L$, what are the allowed wave vectors $k$ for the normal modes in this case?

(d)

Consider a general standing wave

$$y(x,t)=A_o \sin(k x+\phi_o) \cos(\omega t).$$

Does such a wave satisfy your boundary condition in (a) if $b$ has any value other than $0$ or $\infty$? Explain your result for these arbitrary values of $b$ in a short sentence or two.
Hint: For the last part, consider conservation of energy.