Question

if you double the tension on a rope, how much faster do waves travel on it?

and if you triple and quadruple the tension.

Answer 1

the tension in the string and the speed of the wave are related as

$v = \sqrt{\frac{T}{\rho }}$

now if the tension is doubled then

$v' = \sqrt{\frac{2T}{\rho }} = \sqrt{2} v$

i.e. the velocity get increased by factor $\sqrt{2}$

and if it is tripled , quadrupled the velocity get increased by factor $\sqrt{3}$ and 2