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  4. Transverse waves of the same frequency are generated in two steel wires A and B. The diameter of A is twice of B and the tension in A is half that in B. The ratio of the velocities of waves in A and B is

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Q. Transverse waves of the same frequency are generated in two steel wires $A$ and $B$. The diameter of $A$ is twice of $B$ and the tension in $A$ is half that in $B$. The ratio of the velocities of waves in $A$ and $B$ is

KCETKCET 2001Oscillations

Solution:

$\upsilon_{wave} = \sqrt{\frac{T}{m}} = \sqrt{\frac{T}{\rho \frac{\pi d^2}{4}l}}$ i.e. $\upsilon_{wave} \propto \, \sqrt{\frac{T}{d}}$
$\therefore $ $\frac{\upsilon_{wave\,A}}{\upsilon_{wave\,B}} = \sqrt{\frac{T_A}{T_B}} \times \frac{d_B}{d_A} = \sqrt{\frac{\frac{1}{2}T_B}{T_B}} \times \frac{d_B}{2d_A} = \frac{1}{2\sqrt{2}}$