Explanation

(d) Frequency of sonometer wire is given by
$v = \frac{1}{2l}\sqrt{\frac{T}{m}}$
Where m is mass of string per unit length, and T is tension in the string.
Also, m=πr$^{2}$d
R being radius of string per unit length, and T is tension in the string.
So,
$v = \frac{1}{2l}\sqrt{\frac{T}{\pi r^{2}d}}$
Or
$v \propto \frac{\sqrt{T}}{r}$
Or
$\frac{v_{1}}{v_{2}} = \sqrt{\frac{T_{1}}{T_{2}}} \times \left( \frac{r_{2}}{r_{1}} \right)$
$Given,\ r_{2} = 2r_{1},\ \ \ \ \ \ \ \ T_{2} = \frac{T_{1}}{2},\ \ \ \ v_{1} = v$
Hence,
$\frac{v}{v_{2}} = \sqrt{2} \times 2$
Or
$v_{2} = \frac{v}{2\sqrt{2}}$