Tension at a section $x$ distance from free end is

$T=mg=\mu ·x·g$

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

Wave speed over the rope is given by 

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

$\Rightarrow v^2=gx$

Comparing above equation with 

$v^2-u^2=2as$, we get

$\Rightarrow v^2=2ax=gx$

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

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

then from 

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

we have 

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

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

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

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