Transverse and Longitudinal Waves — Question 1

Question 1

Wave type is defined relative to propagation, not the page

A plane mechanical disturbance propagates along $\hat{\mathbf n}=(3\hat{\mathbf i}+4\hat{\mathbf j})/5$. Its displacement is $\mathbf u=a(4\hat{\mathbf i}-3\hat{\mathbf j})\cos\Phi/5$.

1. Classify the displacement as transverse or longitudinal.

2. If the drawing is rotated, can the classification change? Explain with a coordinate-independent test.

Question 1 - Solution

Wave type is defined relative to propagation, not the page

Method and mathematical working

Project displacement onto the unit propagation vector. The longitudinal scalar component vanishes exactly when the motion is perpendicular to propagation.$$\begin{aligned}u_\parallel&=\mathbf u\cdot\hat{\mathbf n},\qquad \mathbf u_\perp=\mathbf u-u_\parallel\hat{\mathbf n}.\\[4pt]\widehat{\mathbf n}\cdot\mathbf u&=\frac{a\cos\Phi}{25}(3\cdot4+4\cdot(-3))=0.\end{aligned}$$

(a) Calculation and conclusion

The dot product is $\mathbf u\cdot\hat{\mathbf n}=a(12-12)\cos\Phi/25=0$, so displacement is transverse. Neither vector has to be horizontal or vertical.

(b) Interpretation and limits

Rigid rotation preserves dot products and angles. Perpendicular displacement remains transverse in every rotated coordinate system. The classification refers to displacement relative to propagation, not relative to the sheet of paper.

Physical check

Both given direction vectors have unit length. Their zero dot product therefore establishes a right angle independently of their visual orientation.