The Speed of a Traveling Wave — Question 1

Question 1

The sign of a constant-phase trajectory fixes direction

A wave is $y=A\sin(6x+18t+\phi)$ in SI units, with $A>0$.

1. Follow a point of constant phase to find the signed propagation velocity.

2. Compare with $y=-A\sin(6x+18t+\phi)$. Does the leading minus sign reverse propagation?

Question 1 - Solution

The sign of a constant-phase trajectory fixes direction

Method and mathematical working

Treat the phase as a constant along the trajectory of a crest or any other identifiable phase. Differentiate this constraint with respect to time; amplitude is outside the constraint.$$\begin{aligned}6x+18t+\phi&=C,\\6\frac{dx}{dt}+18&=0,\qquad \frac{dx}{dt}=-3\,\mathrm{m/s}.\\[4pt]-A\sin\Phi&=A\sin(\Phi+\pi),\qquad \frac{d(\Phi+\pi)}{dt}=6\dot x+18=0.\end{aligned}$$

(a) Calculation and conclusion

Constant phase requires $6x+18t+\phi=C$, so $dx/dt=-18/6=-3\,\mathrm{m/s}$. The wave travels left with speed $3\,\mathrm{m/s}$.

(b) Interpretation and limits

The leading minus sign is a phase shift of $\pi$, since $-\sin\Phi=\sin(\Phi+\pi)$. It changes displacement signs but not the slope of a constant-phase trajectory. Reversing the relative sign of the space and time terms reverses propagation.

Physical check

The dimensions are $(\mathrm{rad/s})/(\mathrm{rad/m})=\mathrm{m/s}$. Multiplying the entire displacement by $-1$ preserves the same trajectory equation and therefore the same direction.