The Speed of a Traveling Wave — Question 3

Question 3

Local slope and local velocity can infer a translation speed

A profile is known to translate rigidly as $y=f(x-ct)$, with signed velocity $c$. At one point $y_x=-.040$ and $y_t=.12\,\mathrm{m/s}$.

1. Infer $c$ and its direction.

2. Why does the method fail at a smooth crest? What condition should several nonzero-slope measurements satisfy if rigid translation is valid?

Question 3 - Solution

Local slope and local velocity can infer a translation speed

Method and mathematical working

Apply the chain rule to the same profile argument in both partial derivatives. The unknown derivative of the profile cancels only where that derivative is nonzero.$$\begin{aligned}y_x&=f^{\prime}(x-ct),\qquad y_t=-c f^{\prime}(x-ct),\\c&=-\frac{y_t}{y_x}=-\frac{0.12}{-0.040}=3.0\,\mathrm{m/s}.\\[4pt]c&=-\frac{0.12\,\mathrm{m/s}}{-0.040}=+3.0\,\mathrm{m/s},\qquad y_x=0\Rightarrow -y_t/y_x=0/0.\end{aligned}$$

(a) Calculation and conclusion

The chain rule gives $y_t=-cy_x$, so $c=-.12/(-.040)=3.0\,\mathrm{m/s}$, to the right. The slope is dimensionless, leaving speed units in the ratio.

(b) Interpretation and limits

At a smooth crest both derivatives vanish, producing an indeterminate $0/0$. At all points with nonzero slope the ratio $-y_t/y_x$ should give the same $c$. Different values signal deformation, superposed directions or measurement error rather than one rigidly translating profile.

Physical check

For a rightward wave, a negative spatial slope must correspond to upward local motion. The measured signs satisfy that condition. A crest requires neighboring measurements because its two first derivatives vanish.