Travelling waves: what moves along, and what only bobs up and down

Travelling waves

When a wave runs along a rope, why does each bit of rope only move up and down?

Shake one end of a rope and watch the wave front advance at v = λf while a marked point waits for it, then only bobs in place. Change the frequency and the wavelength adjusts so the speed stays the same.

Drag the ringed points on the board.

High school
Wavelength, frequency, period
v = λf
Particles oscillate in place
Wave graph y–x

A

1.0

\lambda

4.0

f

0.5

x_P

6.0

t

5.00

v = \lambda f = \dfrac{\lambda}{T}

$v = \lambda f$: 2.00
$T = 1/f$: 2.00
$t_{\text{arrive}} = x_P / v$: 3.00
$y_P$: 0.00

Point Poscillating

Guided tour

1Speed = wavelength × frequency
The source shakes at $f=0.5\,\text{Hz}$ , one full shake every $T=2\,\text{s}$ , and each shake lays down one wavelength, $\lambda=4\,\text{m}$ . So the front moves $4\,\text{m}$ every $2\,\text{s}$ : $v=\lambda f=2\,\text{m/s}$ .
2Shake faster, same rope
The speed is set by the rope, not by your hand. Shake twice as fast, $f=1\,\text{Hz}$ , and the waves bunch up to $\lambda=2\,\text{m}$ ; the front still moves at $2\,\text{m/s}$ .
3P waits for the wave
Back to $\lambda=4\,\text{m}$ , $f=0.5\,\text{Hz}$ , and look at $P$ , $6\,\text{m}$ from the source. At $t=2.5\,\text{s}$ the front is only at $5\,\text{m}$ and $P$ has not moved. The wave arrives at $t=x_P/v=3\,\text{s}$ .
4Up and down, never along
At $t=3.5\,\text{s}$ , a quarter period after the wave arrived, $P$ is at the crest, $y_P=1\,\text{m}$ , directly above where it started. Press play: $P$ repeats the source's motion $3\,\text{s}$ late and never travels with the wave. What travels is the shape, and the energy with it.

Explanation

1
What a wave carries
Each bit of rope is pulled by its neighbour and copies its motion a moment later. That delay is what makes the shape travel. The particles themselves only oscillate about their rest positions; the wave carries the pattern of motion and energy, not matter.
2
v = λf
In one period $T$ the source makes one full oscillation and the front moves one wavelength, so
$v=\frac{\lambda}{T}=\lambda f$
The medium fixes $v$ (a tighter rope is faster, sound in air about $340\,\text{m/s}$ ); the source fixes $f$ ; $\lambda$ follows from both. When a wave enters a new medium its frequency stays the same and its wavelength changes.
3
Every point is the source, delayed
A point at distance $x$ does what the source did $x/v$ seconds earlier:
$y(x,t)=A\sin\left[2\pi\left(ft-\frac x\lambda\right)\right]$
Points one wavelength apart are always in step; points half a wavelength apart always move in opposite directions.

Check yourself

Q01

Sound in air travels at $340\,\text{m/s}$ . A $680\,\text{Hz}$ tone has a wavelength of

Q02

You shake the end of a rope twice as fast. On the same rope,

Q03

As a wave passes, a cork on a pond

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