The velocity of a transverse wave in a stretched string is given by,

\(v=\sqrt{\frac T\mu}\)

Where,

T = Tension on string

μ = \(\frac mL\), mass per unit lengths = \(\frac{\rho\pi d^2}L\)

d = diameter of wire,

So, \(v=\sqrt{\frac{4T}{\rho d^2}}=\;\frac2d\sqrt{\frac T{\rho\pi}}\)

∴ \(v\propto\frac1d\). Thus, the velocity of wave is highest for thinnest string.