Two vibrating strings of the same material but lengths $L$ and $2L$ have radii $2r$ and $r$ respectively. They are stretched under the same tension. Both the strings vibrate in their fundamental modes,the one of length $L$ with frequency $f_1$ and the other with frequency $f_2$. The ratio $\frac{f_1}{f_2}$ is given by

The frequency of the first harmonic of a string stretched between two points is $100 \, Hz$. The frequency of the third overtone is ... $Hz$.

$A$ wire of density $9 \times 10^3 \text{ kg/m}^3$ is stretched between two clamps $1 \text{ m}$ apart and is subjected to an extension of $4.9 \times 10^{-4} \text{ m}$. The lowest frequency of transverse vibration in the wire is ..... $\text{Hz}$ $(Y = 9 \times 10^{10} \text{ N/m}^2)$.

Two identical strings $X$ and $Z$ made of the same material have tensions $T_{x}$ and $T_{z}$ in them. If their fundamental frequencies are $450\, Hz$ and $300\, Hz$,respectively,then the ratio $T_{x} / T_{z}$ is:

Two wires $W_1$ and $W_2$ have the same radius $r$ and respective densities $\rho_1$ and $\rho_2$ such that $\rho_2 = 4\rho_1$. They are joined together at the point $O$,as shown in the figure. The combination is used as a sonometer wire and kept under tension $T$. The point $O$ is midway between the two bridges. When a stationary wave is set up in the composite wire,the joint is found to be a node. The ratio of the number of antinodes formed in $W_1$ to $W_2$ is

If the length of a stretched string is shortened by $40 \%$ and the tension is increased by $44 \%$,then the ratio of the final and initial fundamental frequencies is :