Find the ratio of the fundamental tone frequencies of two identical strings after one of them was stretched by `eta_(1)=2.0 %` and the other, by `eta_(2)=4.0 %`. The tension is assumed to be proportional to the elongation.

Explanation

`n = (1)/(2 l) sqrt((T)/(m))` . If `I_(0)` be the initial length and `f` be the fractional increase in length , `I = I_(0) + fI_(0)`. Since tension is proportional to the increase in length , `T = k xx fI_(0)` where `k` is a constant .
` m = (M)/(I_(0) + fI_(0))`
where `M` is the mass of the string
`:. (1)/(2 I_(0) ( 1 + f)) sqrt (( k f I_(0)( 1 + f))/( M)) = (1)/( 2 I_(0)) sqrt ((kI_(0) f)/( M( 1+ f))`
Since `I_(0) , k and M` are constants
` n prop sqrt ((f)/( 1 + f))`
`:. (n_(1))/( n_(2)) sqrt(( f_(1) ( 1 + f_(2)))/( f_(2)( 1 + f_(1)))) = sqrt ((0.02( 1 + 0.04))/(0.04( 1 + 0.02))) = 0.71`