Wave Travelling Through Two Wires

Problem

A steel wire of length 33.0 m and a copper wire of length 24.0 m, both with 1.00-mm diameters, are connected end to end and stretched to a tension of 170 N. The density of steel is 7860 kg/m³, and the density of copper is 8920 kg/m³. How much time would a transverse wave travel the entire length of the two wires?

Solution

The first step is to calculate the linear mass density for the steel and copper wires. Since $\mu = M /L$ (mass over length), we must calculate the mass using the density multiplied by the volume.

$\mu = \frac{\rho AL}{L} = \rho A$

Here $A$ is the cross-sectional area, which we assume are circles from the problem.

Thus,

$\mu = \rho \left(\frac{\pi}{4}d^2\right)$

while

${\mu}_{steel} = 7860 \left(\frac{\pi}{4}(0.001)^2\right) = 6.173\times{10}^{-3} kg/m$

and

${\mu}_{copper} = 8920 \left(\frac{\pi}{4}(0.001)^2\right) = 7.006\times{10}^{-3} kg/m$.

To calculate the time it would take for the transverse wave to travel the entire length of the two wires, we need to find the travel time of the waves through each wire and add them up.

${t}_{steel} = \frac{{L}_{steel}}{{v}_{steel}}$

${t}_{copper} = \frac{{L}_{copper}}{{v}_{copper}}$

$t = {t}_{steel} + {t}_{copper}$

Since $v = \sqrt{\frac{T}{\mu}}$ for waves travelling across a wire, the speed of the wave through the steel wire is

${v}_{steel} = \sqrt{\frac{170}{6.173\times{10}^{-3}}} = 165.95 m/s$ .

Similarly, the speed of the wave through the copper wire is

${v}_{copper} = \sqrt{\frac{170}{7.006\times{10}^{-3}}} = 155.77 m/s$.

Subsequently,

${t}_{steel} = \frac{33.0}{165.95} = 0.199$

and

${t}_{copper} = \frac{24.0}{155.77} = 0.154$.

Therefore the time it would take for the transverse wave to travel across both wires is

$t = 0.199 + 0.154 = 0.353 s$.