Transverse and Longitudinal Waves — Question 7

Question 7

A transverse pulse gives opposite local motions on its two slopes

A smooth upward pulse moves right at speed $v$ without changing shape: $y(x,t)=f(x-vt)$. At one point the spatial slope is positive; at another it is negative.

1. Relate the local vertical velocity to the spatial slope.

2. State which point moves downward and explain why neither vertical velocity is the horizontal pulse speed.

Question 7 - Solution

A transverse pulse gives opposite local motions on its two slopes

Method and mathematical working

Keep the material position fixed when taking the time derivative. The negative sign arises because a rightward translation reduces the argument $x-vt$ at that position.$$\begin{aligned}\frac{\partial y}{\partial t}&=f^{\prime}(x-vt)(-v),\qquad \frac{\partial y}{\partial x}=f^{\prime}(x-vt).\\[4pt]y_x>0&\Rightarrow y_t=-vy_x<0,\\y_x<0&\Rightarrow y_t>0,\qquad y_x=0\Rightarrow y_t=0.\end{aligned}$$

(a) Calculation and conclusion

The chain rule gives $\partial y/\partial t=-v f^{\prime}(x-vt)=-v\,\partial y/\partial x$. The positive-slope point moves downward and the negative-slope point upward.

(b) Interpretation and limits

The field varies because different parts of the moving profile pass a fixed material label. Its vertical velocity depends on local slope and can vanish at a smooth crest even though the crest continues horizontally at $v$. Direction and magnitude distinguish particle velocity from wave speed.

Physical check

A smooth crest has zero slope and hence zero instantaneous transverse velocity. The crest can nevertheless continue translating horizontally at nonzero speed.