A heavy uniform rope is held vertically by clamping it to a rigid support at the upper end. A wave of a certain frequency is set up at the lower end. The wave will

Due to the weight of the rope, the tension will increase along the string from the lower end to the upper end. Hence, the wave will travel with an increasing velocity along the string.

Tension in the rope at a distance x from the bottom, $T=\frac{Mg}{L}x$

Speed of pulse, $v=\sqrt{\frac{T}{\mu }}$

$\Rightarrow v=\sqrt{\frac{Mgx}{\mu L}}$

$\Rightarrow \mathrm{v}\propto \sqrt{x}$