Two strings A and B , constructed from the identical material, are...

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

The linear mass density can be written as $\mu = \rho \pi r^2$, where $\rho$ is the density of the material and $r$ is the radius of the string.

Since both strings are made of the same material and subjected to the same tension, $T$ and $\rho$ are constant.

Therefore, the speed $v$ is inversely proportional to the radius $r$:

$v \propto \dfrac{1}{r}$

The ratio of the speeds is:

$\dfrac{v_A}{v_B} = \dfrac{r_B}{r_A}$

Given that $r_A = 2r_B$, substituting this value gives:

$\dfrac{v_A}{v_B} = \dfrac{r_B}{2r_B} = \dfrac{1}{2}$