Standing waves

IB Physics SL· Unit 3.4 Standing waves· 15 min read

1. Formation and core properties of standing waves★★☆☆☆⏱ 4 min

📘 Definition

Standing wave

A stationary wave pattern formed when two identical waves interfere after travelling in opposite directions along the same medium

Example:

An incoming wave interfering with its own reflection off a fixed boundary

Unlike travelling waves, standing waves do not transfer net energy through the medium. All points between adjacent nodes oscillate in phase, with amplitude varying from zero (at nodes) to maximum (at antinodes).

Key spacing rules for IB exams:

Distance between adjacent nodes or adjacent antinodes: $\frac{λ}{2}$
Distance between a node and the nearest antinode: $\frac{λ}{4}$

📐 Worked Example

Adjacent antinodes on a standing wave are 20 cm apart. What is the wavelength of the original travelling waves?

1
Recall adjacent antinodes are separated by half a wavelength:
2
$\frac{\lambda}{2} = 20 \text{ cm}$
3
Rearrange to solve for total wavelength:
4
$\lambda = 2 \times 20 = 40 \text{ cm}$

Exam tip:

Always label nodes and antinodes clearly when asked to draw a standing wave in an exam question

2. Standing waves on strings fixed at both ends★★☆☆☆⏱ 4 min

When a string is fixed at both ends, each fixed end cannot move, so both ends must be nodes. This creates a constraint on allowed wavelengths: the length of the string $L$ must equal an integer multiple of half wavelengths.

L = n \frac{\lambda_n}{2}, \quad n = 1, 2, 3, ...

Using the wave equation $v = f λ$ , we can rearrange to get the frequency of the $n$ th harmonic:

f_n = \frac{n v}{2 L}

$n = 1$ is the fundamental (first harmonic), $n = 2$ the second harmonic, and so on. All integer harmonics are allowed for fixed-end strings.

📐 Worked Example

A 1.5 m string fixed at both ends has a wave speed of 300 m/s. Calculate the frequency of the second harmonic.

1
Identify values: $n = 2$ , $v = 300$ m/s, $L = 1.5$ m
2
Substitute into the fixed string frequency formula:
3
$f_2 = \frac{n v}{2 L} = \frac{2 \times 300}{2 \times 1.5}$
4
Calculate the final result:
5
$f_2 = \frac{600}{3} = 200 \text{ Hz}$

3. Standing waves in open and closed air columns★★★☆☆⏱ 4 min

Standing longitudinal sound waves form in air columns (e.g. organ pipes, wind instruments). Boundary rules:

Closed end: air cannot move, so it is a node of displacement
Open end: air moves freely, so it is an antinode of displacement

There are two common air column systems in IB exams:

1. Open at both ends: Both ends are antinodes. All integer harmonics allowed: $f_{n} = \frac{n v}{2 L}, n = 1, 2, 3...$
1. Closed at one end, open at the other: One node, one antinode. Only odd harmonics allowed: $f_{n} = \frac{n v}{4 L}, n = 1, 3, 5...$

📐 Worked Example

A 0.4 m pipe closed at one end and open at the other has a fundamental frequency of 210 Hz. Calculate the speed of sound in the pipe.

1
Fundamental frequency for closed pipe is $n = 1$ , so use the closed pipe formula:
2
$f_1 = \frac{1 \times v}{4 L}$
3
Rearrange to solve for speed $v$ :
4
$v = 4 L f_1 = 4 \times 0.4 \times 210$
5
Calculate the result:
6
$v = 336 \text{ m/s}$

4. Differences between standing and travelling waves★★☆☆☆⏱ 3 min

Property	Standing wave	Travelling wave
Net energy transfer	No net transfer	Net transfer along the medium
Amplitude	Varies from 0 to maximum	Constant for all points
Phase	All points between nodes in phase	Phase changes continuously along the wave
Wavelength definition	Adjacent nodes = $λ /2$	Adjacent peaks = $λ$

✓ Quick check

Test your understanding:

Which property is unique to standing waves, not travelling waves?
- All points have the same amplitude of oscillation
- No net energy transfer along the medium
- Oscillation can be transverse to wave direction
- Wave speed depends on the medium
Reveal answer
1 —
Correct! Travelling waves transfer net energy, while standing waves do not. Transverse oscillation is not unique to standing waves.

5. Common Pitfalls

Wrong move:

Taking node-to-antinode distance as $λ /2$

Why:

Only adjacent nodes or adjacent antinodes are $λ /2$ apart. Node to nearest antinode is half this value

Correct move:

Always remember: node to adjacent antinode distance = $λ /4$

Wrong move:

Using all integer $n$ for closed-end pipe harmonics

Why:

Pipes closed at one end only allow odd harmonics, so even values of $n$ are not valid standing wave patterns

Correct move:

For a pipe closed at one end, only use odd $n$ with the formula $f_{n} = \frac{n v}{4 L}$

Wrong move:

Assuming open ends of air columns are nodes

Why:

Open ends allow air to move freely, so they are antinodes, not nodes. This flips all calculations if you get it wrong

Correct move:

Memorize the rule: fixed/closed boundaries = nodes, free/open boundaries = antinodes

Wrong move:

Forgetting that standing waves can be longitudinal

Why:

Most examples use transverse standing waves on strings, but sound standing waves in air columns are longitudinal, which are very common in exams

Correct move:

Recognize that standing waves can form for any wave type, transverse or longitudinal

6. Quick Reference Cheatsheet

System	Boundary Conditions	Frequency Formula	Allowed Harmonics
String fixed both ends	2 nodes	$f_{n} = \frac{n v}{2 L}$	$n = 1, 2, 3...$
Pipe open both ends	2 antinodes	$f_{n} = \frac{n v}{2 L}$	$n = 1, 2, 3...$
Pipe closed one end	1 node, 1 antinode	$f_{n} = \frac{n v}{4 L}$	$n = 1, 3, 5...$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · Paper 1
Node spacing on a fixed string
2023 · Paper 2
Harmonics in closed air column

What's Next

Standing waves are a core application of the superposition principle, and form the basis for how musical instruments produce sound, as well as resonance problems that frequently appear in both Paper 1 and Paper 2 of IB Physics SL. Mastering boundary conditions for standing waves also builds foundational understanding for more advanced wave topics like double-slit interference and diffraction that you will cover later in the course. This topic also connects closely to sound wave properties tested in subsequent units.