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(D) The fundamental frequency of transverse vibration is given by $f_T = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$,where $T$ is tension,$L$ is length,and $\m

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$1:\sqrt{\eta }$

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(D) The fundamental frequency of transverse vibration is given by $f_T = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$,where $T$ is tension,$L$ is length,and $\mu$ is mass per unit length.The fundamental frequency of longitudinal vibration is given by $f_L = \frac{1}{2L} \sqrt{\frac{Y}{\rho}}$,where $Y$ is Young's modulus and $\rho$ is density.Using $Y = \frac{T/A}{\Delta L/L}$,we have $\frac{Y}{\rho} = \frac{T}{\rho A} \cdot \frac{L}{\Delta L} = \frac{T}{\mu} \cdot \frac{L}{\Delta L}$.Thus,$\frac{f_T}{f_L} = \sqrt{\frac{T/\mu}{Y/\rho}} = \sqrt{\frac{T/\mu}{(T/\mu) \cdot (L/\Delta L)}} = \sqrt{\frac{\Delta L}{L}}$.Given $\frac{\Delta L}{L} = \frac{1}{\eta}$,the ratio is $\sqrt{\frac{1}{\eta}} = \frac{1}{\sqrt{\eta}}$.

$A$ string is stretched so that its length is increased by $\frac{1}{\eta }$ of its original length. The ratio of the fundamental frequency of transverse vibration to that of the fundamental frequency of longitudinal vibration will be

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