Velocity of transverse wave in a stretched string is ‘V’ when elongation produced in it is ‘x’, what will be the velocity of transverse wave in the same string when elongation produced in it is ‘2x’ ?

If Y be the young’s modulus of the material of the string, µ be the mass per unit length and S be the cross-sectional area, Then

$$\begin{gathered} \mathrm{V}=\sqrt{\frac{\mathrm{T}}{\mathrm{\mu }}}\mathrm{and}\mathrm{T}=\mathrm{Area}\times \mathrm{Stress}=\mathrm{S}.\mathrm{\gamma }.\frac{\mathrm{x}}{\mathrm{L}} \\ \therefore \begin{array}{l}\mathrm{V}=\sqrt{\frac{\mathrm{S}.\mathrm{Y}.\frac{\mathrm{x}}{\mathrm{L}}}{\mathrm{\mu }}}\mathrm{and}\mathrm{V}\text{'}=\sqrt{\frac{\mathrm{S}.\mathrm{Y}.\frac{2\mathrm{x}}{\mathrm{L}}}{\mathrm{\mu }}}\\ \therefore \frac{\mathrm{V}\text{'}}{\mathrm{V}}=\sqrt{2}\Rightarrow \mathrm{V}\text{'}=\sqrt{2}.\mathrm{V}\end{array} \end{gathered}$$