Transverse and Longitudinal Waves — Question 3

Question 3

A longitudinal displacement graph is not the shape of a bent medium

A row of particles has equilibrium coordinates $X=0,1,2,3\,\mathrm{m}$. At one instant their longitudinal displacements are $u=0,.10,0,-.10\,\mathrm{m}$ respectively. Actual positions are $x=X+u$.

1. Find the three neighboring separations and identify compressed intervals.

2. If $u$ is plotted vertically against $X$, does the sinusoidal-looking graph mean the particles moved transversely?

Question 3 - Solution

A longitudinal displacement graph is not the shape of a bent medium

Method and mathematical working

Convert each displacement into an actual coordinate before taking neighboring differences. A separation depends on two displacement values, not on one value alone.$$\begin{aligned}x_{j+1}-x_j&=(X_{j+1}-X_j)+(u_{j+1}-u_j).\\[4pt](x_0,x_1,x_2,x_3)&=(0,1.10,2.00,2.90)\,\mathrm m,\\(x_1-x_0,x_2-x_1,x_3-x_2)&=(1.10,0.90,0.90)\,\mathrm m.\end{aligned}$$

(a) Calculation and conclusion

Actual positions are $0,1.10,2.00,2.90\,\mathrm{m}$. Separations are $1.10,.90,.90\,\mathrm{m}$: the last two intervals are compressed relative to the original $1\,\mathrm{m}$ spacing.

(b) Interpretation and limits

No. The vertical graph coordinate represents the numerical value of longitudinal displacement; it is not a second physical position coordinate. The particles still lie on the same line. Axis labels determine the graph meaning.

Physical check

A negative difference $u_{j+1}-u_j$ reduces the spacing even when both particles move in the same direction. The plotting offset between rows is not a transverse material displacement.