Deriving your own wave equation

Take the same string system from class lecture and the lecture notes, but with drag on each chunk of the string contributing an addition force $\vec F=-b m_{\mathrm{ch}} \vec v$.

(a)

Redraw the force-body diagram of Figure 2 from the lecture notes ``Intro to Waves'' including the new force. Explain (briefly, one or two short sentences) why this force does not affect motion in the $x$ direction and hence does not modify Eq. (7) in the notes.

(b)

Following the analysis of Section 4.2.2 of the lecture notes, rewrite in their new form any of Eq. (8), Eq. (11) and Eq. (13) which change due to the presence of the new force.
Hint: You should find in the end that your new equation of motion is equivalent to

$$\frac{\partial^2 y}{\partial t^2}+b \frac{\partial y}{\partial t} = c^2 \frac{\partial^2 y}{\partial x^2},$$

where $c\equiv \sqrt{\tau/\mu}$.

(c)

How quickly will the drag cause the amplitude of a standing wave to decay to $1/e$ of its original amplitude?
Hint: The answer is $1/a$ if the solution to the wave equation has the form

$$y(x,t)=A e^{-at} \cos(\omega t) \cos(k x)$$

To find $a$, assume a solution with the complex representation

$$y(x,t)=\mbox{Re\,}\left( A e^{i\underline{\omega}t} e^{\pm ikx} \right)$$

where $\underline{\omega}$ is a complex frequency, solve for $\underline{\omega}$, and then take $a=\mbox{Im\,}\ \underline{\omega}$.