To find the ratio of the velocities of transverse waves in two strings with different radii but identical materials and tension, consider the following:

The wave velocity $ v $ in a string is given by the formula:

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

where $ T $ is the tension and $ \mu $ is the linear mass density of the string, defined as:

$ \mu = \rho \pi R^2 $

Here, $ \rho $ is the density of the material, and $ R $ is the radius of the string. Given that both strings have the same tension $ T $ and material, we compare their wave velocities $ v_1 $ and $ v_2 $ for radii $ R_1 = R $ and $ R_2 = \frac{R}{2} $.

The velocity ratio is expressed as:

$ \frac{v_2}{v_1} = \frac{\sqrt{\frac{T}{\rho \pi R_2^2}}}{\sqrt{\frac{T}{\rho \pi R_1^2}}} $

Simplifying this expression:

$ \frac{v_2}{v_1} = \frac{\sqrt{R_1^2}}{\sqrt{R_2^2}} = \frac{R_1}{R_2} = \frac{R}{\frac{R}{2}} = 2 $

Thus, the ratio $ \frac{v_2}{v_1} $ is 2.