Answer
$t=2\sqrt {\frac{L}{g}}$
Work Step by Step
The wave speed at a point that is a distance $y$ above the lower end: $v=\sqrt {gy}$
Thus,
at lower point $(y=0)$: $v=0$
and at upper point $(y=L)$: $v=\sqrt {gL}$
Therefore, wave average speed in the string:
$v_{avg}=\frac{1}{2}(0+\sqrt {gL})=\frac{\sqrt {gL}}{2}$
Therefore, the time a transverse wave takes to travel the length of the rope is given by
$t=\frac{L}{v_{avg}}$
or, $t=\frac{L}{\frac{\sqrt {gL}}{2}}$
or, $t=2\sqrt {\frac{L}{g}}$
