Transverse and Longitudinal Waves — Question 9

Question 9

A zero displacement reading cannot classify a wave

At one instant a displacement sensor reports $\mathbf u=\mathbf0$ for a wave traveling along $x$. The sensor can subsequently record all three displacement components.

1. Why can neither transverse nor longitudinal motion be inferred from that one reading?

2. Give a sufficient nonzero time record for each pure type, and state how mixed motion would appear.

Question 9 - Solution

A zero displacement reading cannot classify a wave

Method and mathematical working

A vanishing vector contains no directional information. Classification therefore requires a record with at least one nonzero displacement and sufficient component coverage.$$\begin{aligned}\mathbf u_{\parallel}&=A\sin\omega t\,\mathbf e_x,\quad \mathbf u_{\perp}=A\sin\omega t\,\mathbf e_y,\\\mathbf u_{\parallel}(0)&=\mathbf u_{\perp}(0)=\mathbf0.\end{aligned}$$

(a) Calculation and conclusion

Both a transverse oscillation and a longitudinal oscillation can pass through zero displacement. The zero vector has no measured direction, so this instant cannot determine its relation to propagation.

(b) Interpretation and limits

A purely longitudinal record has $u_y=u_z=0$ at all observed phases and a nonzero varying $u_x$. A purely transverse record has $u_x=0$ with a nonzero perpendicular component. A record with both parallel and perpendicular components indicates mixed motion, assuming measurement covers the relevant phases.

Physical check

Observing only one cycle phase can systematically miss a component. Covering a full cycle avoids mistaking a zero crossing for the absence of that direction of motion.