Stationary Waves

1A rope that stands still

Tie one end of a long skipping rope to a post and shake the other end up and down. At most rates of shaking you get a confused wobble. But at certain rates the rope suddenly settles into a steady pattern of loops: some points of the rope stay completely still while the parts between them swing up and down with a large amplitude, and nothing seems to travel along the rope at all. The waves you send are still travelling; they reflect from the post and come back. What you see is the sum of two progressive waves moving in opposite directions, and it is called a stationary (or standing) wave.

2Formation of a stationary wave

A stationary wave forms when two progressive waves of the same frequency, amplitude and speed travel through the same region in opposite directions. Usually one is a wave and the other is its reflection. By the principle of superposition, the displacement at each point is the sum of the two.

At some instants the two waves line up crest to crest and the string has its largest displacement; a quarter of a period later they cancel everywhere and the string is momentarily straight. At certain fixed points, the nodes, the two waves always cancel, whatever the time.

3The equation of a stationary wave

Write the two waves as

Using $\sin P+\sin Q=2\sin \frac{P+Q}{2}\cos \frac{P-Q}{2}\backslash sin\; P\; +\; \backslash sin\; Q\; =\; 2\backslash sin\backslash frac\{P\; +\; Q\}\{2\}\backslash cos\backslash frac\{P\; -\; Q\}\{2\}$:

This is not of the form $f(x-vt)f(x\; -\; vt)$: nothing travels. Instead, each particle at position $xx$ performs SHM, $\cos \omega t\backslash cos\backslash omega\; t$, with an amplitude $2A\mathrm{\mid }\sin kx\mathrm{\mid }2A|\backslash sin\; kx|$ that depends on where it is.

• Nodes where $\sin kx=0\backslash sin\; kx\; =\; 0$: $kx=n\pi kx\; =\; n\backslash pi$, so $x=0,\frac{\lambda }{2},\lambda ,\frac{3\lambda }{2},\dots x\; =\; 0,\; \backslash frac\{\backslash lambda\}\{2\},\; \backslash lambda,\; \backslash frac\{3\backslash lambda\}\{2\},\; \backslash ldots$. The amplitude is zero.
• Antinodes where $\mathrm{\mid }\sin kx\mathrm{\mid }=1|\backslash sin\; kx|\; =\; 1$: $x=\frac{\lambda }{4},\frac{3\lambda }{4},\dots x\; =\; \backslash frac\{\backslash lambda\}\{4\},\; \backslash frac\{3\backslash lambda\}\{4\},\; \backslash ldots$. The amplitude is $2A2A$.
• Neighbouring nodes (or antinodes) are $\frac{\lambda }{2}\backslash frac\{\backslash lambda\}\{2\}$ apart; a node and the next antinode are $\frac{\lambda }{4}\backslash frac\{\backslash lambda\}\{4\}$ apart.

Phase in a stationary wave

Because the time factor $\cos \omega t\backslash cos\backslash omega\; t$ is the same everywhere, all particles between two neighbouring nodes move in phase: they reach their highest points together. Crossing a node, $\sin kx\backslash sin\; kx$ changes sign, so particles in neighbouring loops move in antiphase. In a progressive wave, by contrast, the phase changes steadily along the wave.

4Experiment 3.1: measuring wave speed from a stationary wave

Aim: to produce stationary waves on a string and use the node spacing to find the wave speed.

Apparatus: a vibration generator and signal generator, a length of string or thin elastic cord, a pulley clamped to the bench, a mass hanger with slotted masses, a metre rule.

Method:

1. Fix the string to the vibration generator, pass it over the pulley and hang about 200 g on it.
2. Slowly raise the frequency from about 10 Hz until a steady pattern of loops appears with large antinodes. Record the frequency $ff$.
3. Measure the distance across several loops and divide to get the node spacing $dd$; then $\lambda =2d\backslash lambda\; =\; 2d$ and $v=f\lambda v\; =\; f\backslash lambda$.
4. Find three or four more patterns at higher frequencies. Plot $\lambda \backslash lambda$ against $\frac{1}{f}\backslash frac\{1\}\{f\}$: the gradient is the wave speed, which should be the same for every pattern.