Types of Waves — Question 3

Question 3

A traveling waveform requires more evidence than repeated oscillation

Two independent pendulums at separated locations are started so that the second lags the first by $.20\,\mathrm{s}$. A rope connects neither pendulum to the other. A student calls their matching delayed records a traveling wave.

1. Explain why the two records alone do not establish propagation.

2. Propose a local perturbation experiment that could distinguish a propagating coupled system from independently timed oscillators.

Question 3 - Solution

A traveling waveform requires more evidence than repeated oscillation

Method and mathematical working

A time delay is a statement about two records. To establish propagation, it must be linked to a mechanism by which a change at one location affects a later state elsewhere.$$\begin{aligned}q_1(t)&=A\cos\omega t,\qquad q_2(t)=A\cos[\omega(t-\tau)].\\[4pt]\phi_2-\phi_1&=-\omega\tau=-0.20\omega,\qquad \delta q_1\ne0\Rightarrow\delta q_2=0\quad\text{when uncoupled}.\end{aligned}$$

(a) Calculation and conclusion

A prescribed phase delay can be created by initial conditions without any disturbance traveling between locations. Wave propagation requires a spatially coupled physical field or medium and a dynamical account of how changes spread; two sinusoidal records do not establish that connection.

(b) Interpretation and limits

Change the motion locally after both oscillators are running and monitor intervening locations and later responses. A coupled medium can transmit the new perturbation with a delay; independent oscillators do not respond to it. Electrical or mechanical common drives must also be excluded as alternate causes.

Physical check

The displayed records can be generated by uncoupled oscillators with different initial phases. A new perturbation after initialization tests transmission rather than the initial timing arrangement.