# Standing waves

> IB Physics SL · Unit 3: Wave Behaviour
> Source: https://www.owlsprep.com/study/ib-physics-sl-u3-standing-waves/

This subtopic covers the formation, key properties and frequency calculations for standing waves on strings and open/closed air columns, a common exam question topic in IB Physics SL.

**Prerequisites:** [Superposition of waves](https://www.owlsprep.com/study/ib-physics-sl-u3-superposition-interference/); [Wave speed, frequency and wavelength](https://www.owlsprep.com/study/ib-physics-sl-u3-wave-characteristics/)

## Learning objectives

- Explain the formation of standing waves via superposition
- Distinguish between standing and travelling waves
- Identify nodes and antinodes for different standing wave systems
- Calculate harmonic frequencies for strings and air columns

## Formation and core properties of standing waves

**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**: $\frac{\lambda}{2}$
- Distance between a node and the nearest antinode: $\frac{\lambda}{4}$

**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

## Standing waves on strings fixed at both ends

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 $v = f\lambda$, 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. $$f_2 = \frac{600}{3} = 200 \text{ Hz}$$

## Standing waves in open and closed air columns

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: $f_n = \frac{n v}{2L}, \quad n=1,2,3...$
- 2. **Closed at one end, open at the other**: One node, one antinode. Only odd harmonics allowed: $f_n = \frac{n v}{4L}, \quad n=1,3,5...$

**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. $$v = 336 \text{ m/s}$$

## Differences between standing and travelling waves

| 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 = $\lambda/2$ | Adjacent peaks = $\lambda$ |

**Check your understanding**

Test your understanding:

1. 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

   *Answer:* No net energy transfer along the medium

   *Why:* Correct! Travelling waves transfer net energy, while standing waves do not. Transverse oscillation is not unique to standing waves.

## Common pitfalls

- **Wrong:** Taking node-to-antinode distance as $\lambda/2$
  - Why it fails: Only adjacent nodes or adjacent antinodes are $\lambda/2$ apart. Node to nearest antinode is half this value
  - Correct: Always remember: node to adjacent antinode distance = $\lambda/4$
- **Wrong:** Using all integer $n$ for closed-end pipe harmonics
  - Why it fails: Pipes closed at one end only allow odd harmonics, so even values of $n$ are not valid standing wave patterns
  - Correct: For a pipe closed at one end, only use odd $n$ with the formula $f_n = \frac{n v}{4L}$
- **Wrong:** Assuming open ends of air columns are nodes
  - Why it fails: Open ends allow air to move freely, so they are antinodes, not nodes. This flips all calculations if you get it wrong
  - Correct: Memorize the rule: fixed/closed boundaries = nodes, free/open boundaries = antinodes
- **Wrong:** Forgetting that standing waves can be longitudinal
  - Why it fails: Most examples use transverse standing waves on strings, but sound standing waves in air columns are longitudinal, which are very common in exams
  - Correct: Recognize that standing waves can form for any wave type, transverse or longitudinal

## Cheatsheet

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

## 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.

- [Fields](https://www.owlsprep.com/study/ib-physics-sl-u4-overview/)
- [Gravitational fields](https://www.owlsprep.com/study/ib-physics-sl-u4-gravitational-fields/)
- [Electric fields](https://www.owlsprep.com/study/ib-physics-sl-u4-electric-fields/)

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