Q.If the tension in sonometer wire is increased by 21%, compare the initial frequency with the later.

The fundamental-frequency formula for a stretched string, $n=\dfrac{1}{2l}\sqrt{T/m}$, splits into three separate proportionality laws, each obtained by holding two of {length l, tension T, linear density m} fixed while the third varies. The LAW OF LENGTH states $n\propto1/l$ when T and m are constant -- a shorter vibrating length gives a higher note. The LAW OF TENSION states $n\propto\sqrt{T}$ when l and m are constant -- tightening a string raises its pitch. The LAW OF LINEAR DENSITY states $n\propto1/\sqrt{m}$ when T and l are constant -- a heavier string of the same length, under the same tension, sounds a lower note, which (writing $m=\pi r^2\rho$ for a string of radius r and material density $\rho$) is equivalently why frequency is inversely proportional to both the string's own radius and the squa …