Learn7408Paper 1 · Paper 1WavesStationary waves

Waves · Subtopic 2 of 6

Stationary Waves

When two identical progressive waves travel in opposite directions, they superpose to form a stationary wave. Nodes (zero amplitude) and antinodes (maximum amplitude) stay in fixed places, $\tfrac\lambda2$ apart. No energy is carried along. On a string fixed at both ends the fundamental has $\lambda=2L$ ; in a pipe closed at one end, $\lambda=4L$ .

Cambridge 9702 Paper 1Cambridge 9702 Paper 2Edexcel YPH11 WPH12AQA 7408 Paper 1≈ 40 minRevision 6 · updated 2026-10-27

Section 1 of 14

Learning objectives

Explain how a stationary wave forms by superposition.
Identify nodes and antinodes and use node spacing $=\tfrac\lambda2$ .
Find the harmonics of strings and air columns.
Measure the wavelength of sound or microwaves using stationary waves.

Pause and think

A guitar string is fixed at both ends. When plucked, which wavelengths can it vibrate with, and why not any wavelength?

Section 2 of 14

Core idea

Start hereA guitar string

Pluck a guitar string and waves race to each end, reflect, and overlap with themselves. Only certain wavelengths fit exactly, producing a pattern that seems to stand still. Those wavelengths decide the notes the string can play.

The big idea

Two equal waves in opposite directions → stationary wave: nodes and antinodes; adjacent nodes $\frac\lambda2$ apart; no net energy transfer. Strings: $L=\frac{n\lambda}2$ . Closed pipe: odd quarter-wavelengths.

1. Formation

A stationary wave on a string: the shape oscillates, nodes stay still

time = 0.0

Between two nodes every point moves in phase; on opposite sides of a node they are in antiphase.

Between adjacent nodes, all points oscillate in phase.
Amplitude varies with position.

2. Strings

Allowed frequencies on a string

$L=n\frac\lambda2$
Nodes at both fixed ends: a whole number of half-wavelengths.
$f_n=\frac v{\lambda_n}=\frac{nv}{2L}$
$v=f\lambda$ .

Worked exampleA 0.80 m string

Wave speed 120 m s $^{-1}$ .

Goal: Fundamental and harmonics.
Strategy: $n=1$ , then multiples.

$\lambda=2L=1.6\text{ m}$
Fundamental.
$f_1=\frac{120}{1.6}=75\text{ Hz}$
$f=v/\lambda$ .
$150,\ 225,\ldots\text{ Hz}$
All multiples.

Answer

75

Hz,

150

Hz,

225

Hz, …

✓ Check

\frac{nv}{2L}

✓.

3. Air columns

Worked exampleResonance tube, 500 Hz fork

Resonances at 0.16 m and 0.50 m.

Goal: Speed of sound.
Strategy: Successive resonances differ by $\frac\lambda2$ .

$0.50-0.16=0.34=\frac\lambda2$
Subtract: end correction cancels.
$\lambda=0.68\text{ m}$
Double.
$v=500\times0.68=340\text{ m s}^{-1}$
$f\lambda$ .

Answer

340

m s

^{-1}

✓ CheckFirst length

0.16\approx\frac\lambda4=0.17

: end correction

0.01

m ✓.

Common misconceptionNode spacing Adjacent nodes are

\frac\lambda2

apart, not

\lambda

Pause and think

Closed pipe: which harmonics?

Summary

Nodes $\lambda/2$ apart.
Strings $L=n\lambda/2$ ; closed pipes odd quarters.

Node: Zero amplitude.
Antinode: Maximum amplitude.

Third harmonic on a string fixed at both ends

Nodes are half a wavelength apart. Here

L=\tfrac32\lambda

, so

\lambda=0.80

m. Points between neighbouring nodes move in phase.

HistoryChladni's singing plates. In

1787

Ernst Chladni sprinkled sand on metal plates and bowed their edges. The sand collected along the nodal lines, forming beautiful patterns. Napoleon was so impressed he paid for Chladni's book to be translated into French.

Fun factThe Tacoma Narrows Bridge. In

1940

wind set up a twisting stationary-wave pattern in the Tacoma Narrows Bridge in the USA. A line down the middle of the deck stayed still (a node) while the edges heaved up and down by metres, until the bridge tore itself apart.

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