The fundamental frequency of vibration \(f\) is given by \(f=\frac{1}{2 l} \sqrt{\frac{T}{\mu}}\)
where \(l=\) length of the wire
\(T=\) tension
= mass per unit length
Thus;
\(f \alpha \frac{1}{l} \ldots . . .( i )\)
\(f \alpha \sqrt{T} \ldots \ldots\).(ii)
or \(f^{2} \alpha T\)
and \(f \alpha \frac{1}{\sqrt{\mu}} \ldots\)....(iii)
(ii) above is a direct relationship. It means increment in \(f\) such as doubling it will result in a geometrical increase in \(T\) i.e. \(T\) quadruples and increasing \(f\) by 4 will increase \(T\) by 16 et cetera