A sonometer wire is vibrating in resonance with a tuning fork. Keeping the tension applied same, the length of the wire is doubled. Under what conditions would the tuning fork still be is resonance with the wire?

Explanation

Initially `f _(1) = (1)/( 2L ) sqrt ((T)/(mu)) = f `
(Frequency of given tuning fork)
Finally, `f ._(1) = (1)/(2 (2 L )) sqrt((T)/(mu))`
`(because ` Length of wire is made doubled)
For second harmonic of sonometer wire,
`2 f._(1) = 2 xx (1)/(2 (2L)) sqrt ((T)/( mu ))`
`= (1)/( 2 L ) sqrt ((T)/( mu))= f`
Thus, given tuning form will resonate with given sonometer.
(i) initially in first harmonic (i.e. fundamental frequency).
(ii) finally (when length of wire is doubled in second harmonic (i.e. first overtone)