Types of Waves — Question 1

Question 1

A delayed disturbance differs from a moving piece of matter

Two marks fixed to a rope are $3.0\,\mathrm{m}$ apart. A pulse reaches the second mark $.60\,\mathrm{s}$ after the first. Each mark moves only vertically and returns to its original height.

1. Find the pulse speed and distinguish it from the motion of either mark.

2. Does return of the marks to their original positions prove that no energy crossed the interval? Explain using the forces between neighboring rope elements.

Question 1 - Solution

A delayed disturbance differs from a moving piece of matter

Method and mathematical working

Track the same pulse feature between the two marked material locations. Its horizontal travel time is distinct from the vertical trajectory of either mark.$$\begin{aligned}v_{\mathrm{pattern}}&=\frac{x_Q-x_P}{t_Q-t_P},\qquad \Delta x_{\mathrm{material}}=0.\\[4pt]v&=\frac{3.0\,\mathrm m}{0.60\,\mathrm s}=5.0\,\mathrm{m/s},\qquad x_{P,Q}(t)=\text{constant}.\end{aligned}$$

(a) Calculation and conclusion

The disturbance travels at $3.0/.60=5.0\,\mathrm{m/s}$ along the rope. This is the speed of a pattern, not a horizontal speed of the marked material. A mark has time-dependent vertical velocity as the pulse passes.

(b) Interpretation and limits

No. Neighboring elements exert tension forces and do work on one another while moving. Energy can pass along the rope even when each material element has zero net displacement after the pulse. Net material transport and energy transport are different observations.

Physical check

At each mark, the final displacement can equal the initial displacement while the tension force does nonzero work during the passage. Returning to the initial position is not a measurement of energy flux.