Transverse and Longitudinal Waves — Question 2

Question 2

An oblique displacement contains two components

A wave propagates in the positive $x$ direction and has displacement $\mathbf u=(3\hat{\mathbf i}+4\hat{\mathbf j})a\cos\Phi$.

1. Find its longitudinal and transverse displacement amplitudes and its total vector amplitude.

2. Can it be called purely transverse because its vertical component is larger? Explain what a sensor measuring only $u_y$ would miss.

Question 2 - Solution

An oblique displacement contains two components

Method and mathematical working

For propagation along $x$, use the horizontal coefficient for longitudinal motion and the vertical coefficient for transverse motion. Combine orthogonal components with the Euclidean norm.$$\begin{aligned}\mathbf u&=u_x\hat{\mathbf i}+u_y\hat{\mathbf j},\qquad |\mathbf u|=\sqrt{u_x^2+u_y^2}.\\[4pt]u_{\parallel,\max}&=3a,\qquad u_{\perp,\max}=4a,\qquad |\mathbf u|_{\max}=\sqrt{9+16}\,a=5a.\end{aligned}$$

(a) Calculation and conclusion

The parallel amplitude is $3a$, the perpendicular amplitude is $4a$, and the total is $5a$. Orthogonal projection separates the components without changing the propagation direction.

(b) Interpretation and limits

It is mixed, since the parallel component is nonzero. A vertical-only sensor would detect a transverse component but would not exclude simultaneous longitudinal motion. Full classification needs enough measured components to reconstruct displacement.

Physical check

A component sensor does not measure the missing orthogonal component. No conclusion that this component is absent follows from a larger vertical reading.