1. For a transverse wave on a stretched string,

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

where:

• $T$ = tension in the string
• $\mu$ = mass per unit length

2. The two strings are made of the same material and have the same tension, so $T$ is same for both.

Thus,

$v \propto \frac{1}{\sqrt{\mu}}$

3. Since both strings have circular cross section and same material,

$\mu = \rho A$

where $\rho$ is density of material and $A$ is cross-sectional area.

For a circular cross section,

$A = \pi r^2$

Hence,

$\mu \propto r^2$

4. For the first string:

$r_1 = R \implies \mu_1 \propto R^2$

For the second string:

$r_2 = \frac{R}{2} \implies \mu_2 \propto \left(\frac{R}{2}\right)^2 = \frac{R^2}{4}$

So,

$\mu_2 = \frac{\mu_1}{4}$

5. Now,

$\frac{v_2}{v_1} = \sqrt{\frac{T/\mu_2}{T/\mu_1}} = \sqrt{\frac{\mu_1}{\mu_2}}$

Substitute $\mu_2 = \mu_1/4$:

$\frac{v_2}{v_1} = \sqrt{\frac{\mu_1}{\mu_1/4}} = \sqrt{4} = 2$

6. Therefore, the correct option is:

$\boxed{2}$

So, Option C is correct.