Two uniform strings A and B made of the same material have lengths...

The speed of a transverse wave in 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 expressed as $\mu = \rho A = \rho \dfrac{\pi D^2}{4}$, where $\rho$ is the density of the material and $D$ is the diameter of the string.

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

Thus, the speed of the transverse wave is inversely proportional to the diameter of the string, $v \propto \dfrac{1}{D}$.

The ratio of the speeds of transverse waves in string A to that in string B is:

$\dfrac{v_A}{v_B} = \dfrac{D_B}{D_A}$

Given that the ratio of diameters is $D_A : D_B = 2 : 1$, we get:

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

The lengths of the strings do not affect the speed of the transverse waves.

Answer: $1:2$