Standing waves

What is it about?

A wave travelling towards the end of a rope is reflected there. The incident wave and the reflected wave have the same frequency and wavelength $\lambda = \frac{c}{f}$ and superpose to form a standing wave: some points never move (nodes N), others in between oscillate particularly strongly (antinodes A).

• Fixed end: reflection with a phase jump of $\pi$, there is always a node at the end: $y = 2\hat y \cdot \sin\!\left(2\pi \frac{d}{\lambda}\right) \cdot \cos(\omega t)$, nodes at $d = k \cdot \frac{\lambda}{2}$ from the end.
• Free end: reflection without a phase jump, an antinode at the end.

Neighbouring nodes are always $\frac{\lambda}{2}$ apart.

Normal modes of a medium of length $L$: with equal ends $L = n \cdot \frac{\lambda}{2}$, so $f_n = n \cdot \frac{c}{2L}$; with unequal ends $L = (2n-1) \cdot \frac{\lambda}{4}$, so $f_n = (2n-1) \cdot \frac{c}{4L}$ (odd multiples only). A guitar string with $L = 0.65\,\text{m}$ and $c = 286\,\frac{\text{m}}{\text{s}}$ has $f_1 = 220\,\text{Hz}$, $f_2 = 440\,\text{Hz}$, …

In an air column a closed end is a velocity node, an open end a velocity antinode; the pressure varies the other way round. A stopped pipe only has odd multiples of its fundamental.

A driver with any frequency only produces a large standing wave at the natural frequencies (resonance, Melde’s experiment).

The motion is shown greatly slowed down and enlarged; damping is neglected except for the driven rope, and the end correction of pipes is ignored.

Try it

1. Press “Send the wave again”. When does the pattern lock in?
2. Drag $P$ onto a node and look at the phasors.
3. Compare with a free end. Free end
4. Increase $n$ for the string and count nodes and antinodes. String, n = 4
5. Make one end free and tap the natural frequencies. Fixed–free
6. Sweep the driving frequency through the resonance curve, with sound. Driver

Tasks

Task 1: Guitar string

$L = 0.65\,\text{m}$, both ends fixed, $c = 286\,\frac{\text{m}}{\text{s}}$. Find wavelength and frequency of the fundamental and the first two overtones. Load task

Task 2: Open or stopped?

Two pipes of length $0.5\,\text{m}$ ($c = 340\,\frac{\text{m}}{\text{s}}$), one open, one stopped. Find the first three natural frequencies of each. Load task

Task 3: Melde’s experiment

$L = 1.2\,\text{m}$, $c = 240\,\frac{\text{m}}{\text{s}}$. At which driving frequencies is there resonance? Load task

Ideas for the classroom

• Use: upper secondary, mechanical waves, after reflection and interference; pairs well with a rope wave, Melde’s experiment and Kundt’s tube.
• Applications: string instruments, flutes and organ pipes; “Play sound” lets students hear the natural frequencies.
• Common stumbling blocks:
  • “There is no wave at a node.” – Both waves pass through and cancel there at every instant.
  • “Two nodes are $\lambda$ apart.” – They are $\frac{\lambda}{2}$ apart.
  • “At a node of the air column the pressure is zero.” – At a velocity node the pressure varies most.