$T=m g=\mu \cdot x \cdot g$

where, $\mu=$ mass per unit length of rope.

Wave speed over the rope is given by

$v=\sqrt{\frac{T}{\mu}} =\sqrt{\frac{\mu \cdot x \cdot g}{\mu}}=\sqrt{g \cdot x}$

$\Rightarrow \quad v^{2} =g x$

Comparing above equation with

$v^{2}-u^{2}=2 a s, \text { we get }$

$\Rightarrow v^{2}=2 a x=g x$

$\Rightarrow \text { Acceleration, } a=\frac{g}{2}$

Now, if $t=$ time for wave pulse to cover a distance $x$,

then from

$s=u t+\frac{1}{2} a t^{2},$

we have

$x=\frac{1}{2} \times \frac{g}{2} \times t^{2}$

$\text { or } t =2 \sqrt{\frac{x}{g}}$

$\Rightarrow t \propto \sqrt{x}$

or time to travel complete length of rope $t \propto \sqrt{L} .$