Two uniform wires of the same material are vibrating under the same tension. If the first overtone of the first wire is equal to the second overtone of the second wire and the radius of the first wire is twice the radius of the second wire,then the ratio of the lengths of the first wire to the second wire is:

When a sonometer wire vibrates in the third overtone,there are:

The length and diameter of a metal wire used in a sonometer are both doubled. The fundamental frequency will change from $n$ to:

$A$ wire of length $L$, diameter $d$, and density of material $\rho$ is under tension $T$ and has a fundamental frequency of vibration $n_A$. Another wire of length $2L$, diameter $3d$, and density of material $2\rho$ is vibrated under tension $2T$, and its fundamental frequency of vibration becomes $n_B$. The ratio $n_B : n_A$ is

$A$ steel wire of length $1 \ m$ and mass $0.1 \ kg$ and having a uniform cross-sectional area of $10^{-6} \ m^2$ is rigidly fixed at both ends. The temperature of the wire is lowered by $20^{\circ} C$. If the wire is vibrating in its fundamental mode,find the frequency (in $Hz$).
$(Y_{\text{steel}} = 2 \times 10^{11} \ N/m^2, \alpha_{\text{steel}} = 1.21 \times 10^{-5} /^{\circ} C)$

Two wires made up of the same material are of equal lengths but their radii are in the ratio $1 : 2$. On stretching each of these two strings by the same tension,the ratio between the fundamental frequencies is