Answer
$v = \sqrt{gy}$
Work Step by Step
We can find an expression for the tension in the rope at a point that is a distance $y$ above the lower end:
$\tau = (\frac{y}{L}) (mg)$
$\tau = \frac{ymg}{L}$
We can find the wave speed at a point that is a distance $y$ above the lower end:
$v = \sqrt{\frac{\tau}{\mu}}$
$v = \sqrt{\frac{\frac{ymg}{L}}{m/L}}$
$v = \sqrt{gy}$
