The speed of the transverse wave on the stretched string under...

The speed of a transverse wave on a stretched string is given by $v = \sqrt{\dfrac{T}{\mu}}$, where $T$ is the tension and $\mu$ is the linear mass density.

For the initial state, the tension is $T_1$, so the speed is:

$v_1 = \sqrt{\dfrac{T_1}{\mu}}$

The tension is increased by $T_1$, so the new tension is $T_2 = T_1 + T_1 = 2T_1$.

The new speed is:

$v_2 = \sqrt{\dfrac{2T_1}{\mu}} = \sqrt{2} \sqrt{\dfrac{T_1}{\mu}} = \sqrt{2} v_1$

Comparing this with $v_2 = x v_1$, we get $x = \sqrt{2}$.

Answer: $\sqrt{2}$