Types of Waves — Question 4

Question 4

A single pulse need not possess one wavelength

A rope carries one isolated triangular pulse of width $w=.40\,\mathrm{m}$ moving at $v=2.0\,\mathrm{m/s}$ without changing shape. There is no repeating train of pulses.

1. Find the duration for which a fixed point is disturbed.

2. Explain why labeling $w$ as the wavelength and $v/w$ as an intrinsic frequency is unjustified.

Question 4 - Solution

A single pulse need not possess one wavelength

Method and mathematical working

The detector begins responding when the leading edge reaches it and stops when the trailing edge passes. The relevant distance is the pulse width, not the distance between periodic repetitions.$$\begin{aligned}t_{\mathrm{trailing}}-t_{\mathrm{leading}}&=\frac{x_{\mathrm{leading}}-x_{\mathrm{trailing}}}{v}=\frac wv.\\[4pt]\Delta t&=\frac{0.40\,\mathrm m}{2.0\,\mathrm{m/s}}=0.20\,\mathrm s,\qquad \Delta t\ne T\ \text{for an isolated pulse}.\end{aligned}$$

(a) Calculation and conclusion

The leading and trailing edges are separated by $w$, so the duration at a fixed point is $w/v=.20\,\mathrm{s}$. This is a passage duration, not a repetition period.

(b) Interpretation and limits

A wavelength is the spatial repeat distance of a periodic pattern; this isolated pulse has none. Likewise, no repeating temporal cycle defines a unique frequency. The scale $v/w=5\,\mathrm{s^{-1}}$ is an inverse duration, and a spectral description would involve multiple frequencies.

Physical check

Doubling the pulse width at fixed speed doubles the passage duration. This limiting comparison works without assigning a period or wavelength to an isolated pulse.