C.4 Standing waves and resonance

IB Physics HL· 45 min read

1. Formation of Standing Waves★★☆☆☆⏱ 10 min

📘 Definition

Standing (stationary) wave

A wave formed by superposition of two identical progressive waves traveling in opposite directions, resulting in fixed positions of zero and maximum amplitude, with no net energy transfer.

Example:

An incident wave reflected off a fixed boundary interfering with the incoming wave

Key features of standing waves: nodes are fixed points of zero amplitude, and antinodes are fixed points of maximum amplitude. The distance between two adjacent nodes is $\frac{λ}{2}$ , and the distance between a node and its adjacent antinode is $\frac{λ}{4}$ .

📐 Worked Example

Adjacent antinodes of a standing wave are 12.5 cm apart. What is the wavelength of the original progressive waves?

1
Adjacent antinodes, like adjacent nodes, are separated by half a wavelength:
2
$\frac{\lambda}{2} = 12.5 \text{ cm}$
3
Solve for wavelength:
4
$\lambda = 2 \times 12.5 = 25 \text{ cm} = 0.25 \text{ m}$

Exam tip:

Never confuse standing waves with progressive waves: standing waves do not propagate energy or waveform.

2. Standing Waves on Fixed Strings★★☆☆☆⏱ 12 min

📘 Definition

First harmonic (fundamental frequency)

f_1

The lowest possible resonant frequency of a system, corresponding to the longest possible wavelength that fits the boundary conditions.

A string fixed at both ends has nodes at both ends. This means the length of the string $L$ is always an integer multiple of half wavelengths, so: $L = n \frac{λ _{n}}{2}$ where $n = 1, 2, 3...$ is the harmonic number. Rearranging gives the frequency formula:

f_n = \frac{nv}{2L}

Where $v$ is the speed of the wave on the string.

📐 Worked Example

A 1.5 m string fixed at both ends has a wave speed of 300 m/s. Calculate the frequency of the 2nd harmonic.

1
Use the nth harmonic formula for fixed strings:
2
$f_n = \frac{nv}{2L}$
3
Substitute $n = 2$ , $v = 300$ m/s, $L = 1.5$ m:
4
$f_2 = \frac{2 \times 300}{2 \times 1.5} = 200 \text{ Hz}$

3. Standing Waves in Open and Closed Pipes★★★☆☆⏱ 15 min

Standing waves form in air columns (pipes) with different boundary conditions depending on whether the end is open or closed:

Closed end: Air cannot move, so this is a node
Open end: Air can move freely, so this is an antinode

For pipes:

Open-open (both ends open): All harmonics exist, formula is the same as fixed strings: $f_{n} = \frac{n v}{2 L}, n = 1, 2, 3...$
Closed-open (one end closed, one open): Only odd harmonics exist, formula is $f_{n} = \frac{n v}{4 L}, n = 1, 3, 5...$

📐 Worked Example

A 0.75 m pipe is closed at one end, open at the other. Speed of sound is 340 m/s. Calculate the fundamental frequency.

1
For closed-open fundamental, $n = 1$ , so $L = \frac{λ}{4}$ :
2
$\lambda = 4L = 4 \times 0.75 = 3.0 \text{ m}$
3
Calculate frequency using $f = \frac{v}{λ}$ :
4
$f_1 = \frac{340}{3.0} \approx 113 \text{ Hz}$

4. Resonance★★★☆☆⏱ 10 min

📘 Definition

Resonance

A phenomenon where a system oscillates at maximum amplitude when an external driving force matches the system's natural resonant frequency.

Resonance occurs when the driving frequency matches one of the natural harmonic frequencies of a system (string, air column, etc). This is the working principle behind all acoustic musical instruments, where resonance amplifies the sound at specific harmonic frequencies.

📐 Worked Example

A 256 Hz tuning fork produces the first resonance in a closed pipe when the pipe length is 32 cm. Calculate the speed of sound.

1
First resonance for closed pipe: $L = \frac{λ}{4}$
2
$\lambda = 4 \times 0.32 = 1.28 \text{ m}$
3
Use $v = f λ$ :
4
$v = 256 \times 1.28 = 327.68 \approx 330 \text{ m/s}$

5. Common Pitfalls

Wrong move:

Assuming closed-open pipes have even harmonics

Why:

The node-antinode boundary condition only allows odd multiples of the fundamental wavelength

Correct move:

Only use odd values of $n$ (1, 3, 5...) for closed-open pipes

Wrong move:

Claiming standing waves transfer net energy

Why:

Confused standing wave properties with progressive waves

Correct move:

Remember standing waves do not transfer net energy, energy is stored in nodes and antinodes

Wrong move:

Treating an open pipe end as a node

Why:

Mixed up boundary conditions for open vs closed ends

Correct move:

Always remember: open ends are antinodes, closed ends are nodes

Wrong move:

Using $λ = L$ for the fundamental frequency of a fixed string

Why:

Forgot that both ends are nodes, so only half a wavelength fits

Correct move:

For fixed string fundamental, $λ = 2 L$

6. Quick Reference Cheatsheet

System	Boundary	Allowed $n$	Frequency formula
String fixed both ends	Node-node	1, 2, 3...	$f_{n} = \frac{n v}{2 L}$
Pipe open both ends	Antinode-antinode	1, 2, 3...	$f_{n} = \frac{n v}{2 L}$
Pipe closed one end	Node-antinode	1, 3, 5...	$f_{n} = \frac{n v}{4 L}$

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.

2025 · Paper 1
Harmonic frequency calculation
2024 · Paper 2
Resonance in closed pipe
2023 · Paper 1
Standing vs progressive wave comparison

What's Next

Standing waves and resonance underpin the behavior of nearly all acoustic musical instruments, and are foundational for understanding wave phenomena across all areas of physics, from mechanical sound waves to electromagnetic standing waves in circuits and quantum mechanical matter waves. Mastery of boundary conditions and harmonic frequency calculations is frequently tested in both Paper 1 and Paper 2 IB Physics HL exams, often combined with superposition or wave speed concepts. Understanding resonance also helps explain real-world phenomena like structural resonance in bridges and resonant sound production.