The speed of a transverse wave on a stretched string is given by the formula $v = \sqrt{\frac{T}{\mu}}$, where $T$ is the tension in the string and $\mu$ is its linear mass density. Let the initial tension be $T_1 = T$, then the initial speed is $v_1 = \sqrt{\frac{T}{\mu}}$. If the tension is doubled, the new tension becomes $T_2 = 2T$. The final speed is then $v_2 = \sqrt{\frac{2T}{\mu}} = \sqrt{2} \sqrt{\frac{T}{\mu}} = \sqrt{2} v_1$. The ratio of the initial speed to the final speed is $\frac{v_1}{v_2} = \frac{1}{\sqrt{2}}$, which can be written as $1:\sqrt{2}$.