Waves

What you will understand

Read the units and coefficients in a sinusoidal travelling-wave equation. Distinguish movement of the wave pattern from motion of a material point, then compare points at the same instant.

Quick answer: specify the convention

For $y=A\sin(kx-\omega t+\phi)$ with positive $k$ and $\omega$, constant phase requires $x$ to increase as $t$ increases: the pattern travels toward $+x$ with speed $\omega/k$. With $kx+\omega t$, it travels toward $-x$. Here $k=2\pi/\lambda$ and $\omega=2\pi f$. A material point on the ideal transverse string oscillates in $y$; it does not follow the crest along the string.

Follow the pattern and one material point

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Phase depends on the event being compared

For this one wave, the signed change from event 1 to event 2 is $\Delta\theta=k\Delta x-\omega\Delta t$. At the same time set $\Delta t=0$. At the same position set $\Delta x=0$. “In phase” means the phase difference is an integer multiple of $2\pi$. Equal displacement at one instant is insufficient: the two points may be on different parts of the sine cycle.

Compare space and time separately

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Common mistakes to catch

The coefficient $k$ is not $1/\lambda$, and $\omega$ is not frequency in hertz. Include the factors of $2\pi$. The speed $\omega/k$ describes propagation; $A\omega$ is the maximum speed of a material point in this sinusoidal model. In a transverse string wave the latter direction is perpendicular to propagation. A snapshot against position is different from a displacement–time graph at one position.

Try it yourself 1: identify the speed

For the worked wave, which is the pattern speed? A. $10\,\mathrm{m\,s^{-1}}$; B. $0.80\pi\,\mathrm{m\,s^{-1}}$; C. $20\,\mathrm{m\,s^{-1}}$.

Try it yourself 2: compare at the same instant

Two points of the worked wave are separated by $+\lambda/4$ in the $x$ direction at the same time. What is the signed phase change? A. $0$; B. $+\pi/2$; C. $-\pi/2$.

A useful check on your next visit

Draw a snapshot and a displacement–time trace for one material point. Label the horizontal axis on each. Recover $\lambda$, $f$, pattern speed and maximum particle speed from the worked equation, naming the units before checking your answer.

Choose your next chapter task

Explain the mistake behind any self-check you missed, then use chapter practice to test the corrected idea.

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Sources and review

Technical review: Codex (automated). Sources checked on 23 October 2026. The diagrams, worked numerical models and self-checks are original teaching material. Human subject review is not claimed.