AQA AQA 2019-style practice: A string of length 0.75 m is fixed at both ends and carries waves travelling at 300 m s^-1. Calculate the frequency of the first harmonic and the frequency of the third harmonic.

The harmonic frequencies are f_n = nv2L. First harmonic (n = 1): f_1 = 1 × 3002 × 0.75 = 3001.5 = 200 Hz. Third harmonic (n = 3): f_3 = 3 f_1 = 3 × 200 = 600 Hz (or directly, 3 × 3001.5). Markers reward using f_n = nv2L for the first harmonic and recognising the third harmonic is three times the first.

EnglandPhysicsSyllabus dot point

How do two progressive waves combine to form a standing wave, and why does it have fixed nodes and antinodes?

The formation of stationary waves from two progressive waves travelling in opposite directions, nodes and antinodes, the differences between stationary and progressive waves, and resonance on strings and in air columns.

A focused answer to AQA A-Level Physics 3.3.1.3, covering how stationary waves form from two progressive waves, nodes and antinodes, the differences between stationary and progressive waves, and the harmonics of a string fixed at both ends.

Generated by Claude Opus 4.810 min answerUpdated 2026-06-02

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What this dot point is asking
How a stationary wave forms
Nodes and antinodes
Stationary versus progressive waves
Resonance on a string
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What this dot point is asking

AQA specification point 3.3.1.3 wants you to explain how a stationary wave forms from two progressive waves travelling in opposite directions, describe nodes and antinodes, contrast stationary and progressive waves, and find the harmonic frequencies of a string fixed at both ends.

How a stationary wave forms

Nodes and antinodes

This fixed spacing is the key to finding wavelengths experimentally: measuring the distance between several nodes and dividing gives a reliable value for $\dfrac{\lambda}{2}$ .

Stationary versus progressive waves

Feature	Stationary wave	Progressive wave
Energy	Stored, not transferred	Transferred along the wave
Amplitude	Varies from zero (node) to maximum (antinode)	Same for all points
Phase	Points between two nodes are in phase	Phase varies continuously along the wave

These differences are a common exam comparison: the most reliable points to make are the energy (stored versus transferred) and the amplitude (varying versus constant).

Resonance on a string

The fixed ends must be nodes, which is why only certain wavelengths (and so frequencies) fit; these are the resonant frequencies or harmonics. The first harmonic (fundamental) fits half a wavelength into the length, the second harmonic fits a full wavelength, and so on. The harmonics are integer multiples of the fundamental, which is the basis of musical pitch on stringed instruments. The wave speed on the string is set by the tension and the mass per unit length through $v = \sqrt{\dfrac{T}{\mu}}$ , so tightening a string raises its frequencies, which is how an instrument is tuned.

Try this

Q1. State the distance between adjacent nodes in terms of wavelength. [1 mark]

Cue. Half a wavelength, $\dfrac{\lambda}{2}$ .

Q2. Explain how a stationary wave is formed on a string fixed at both ends. [3 marks]

Cue. A wave reflects at the fixed end and superposes with the incoming wave of the same frequency travelling the other way, producing fixed nodes and antinodes.

Q3. State the relationship between the third harmonic frequency and the first harmonic frequency of a string. [1 mark]

Cue. The third harmonic is three times the first harmonic.

Exam-style practice questions

Practice questions written in the style of AQA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

AQA 20194 marksA string of length

0.75 \text{ m}

is fixed at both ends and carries waves travelling at

300 \text{ m s}^{-1}

. Calculate the frequency of the first harmonic and the frequency of the third harmonic.

Show worked answer →

The harmonic frequencies are $f_n = \dfrac{nv}{2L}$ .

First harmonic ( $n = 1$ ): $f_1 = \dfrac{1 \times 300}{2 \times 0.75} = \dfrac{300}{1.5} = 200 \text{ Hz}$ .

Third harmonic ( $n = 3$ ): $f_3 = 3 f_1 = 3 \times 200 = 600 \text{ Hz}$ (or directly, $\dfrac{3 \times 300}{1.5}$ ).

Markers reward using $f_n = \dfrac{nv}{2L}$ for the first harmonic and recognising the third harmonic is three times the first.

AQA 20214 marksDescribe how a stationary wave is formed on a stretched string fixed at both ends, and state two ways in which a stationary wave differs from a progressive wave.

Show worked answer →

A wave travels along the string and reflects at the fixed end, sending a wave of the same frequency back in the opposite direction. The incident and reflected waves superpose. Where they are always in antiphase the displacements cancel (a node), and where they are always in phase they reinforce (an antinode), giving a stable pattern of nodes and antinodes.

Two differences: a stationary wave stores energy rather than transferring it along the string, whereas a progressive wave transfers energy; and the amplitude of a stationary wave varies with position (zero at nodes, maximum at antinodes), whereas all points on a progressive wave have the same amplitude.

Markers reward the reflection and superposition of two waves of equal frequency, the formation of nodes and antinodes, and two valid differences (energy, amplitude or phase).

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Sources & how we know this

AQA A-level Physics (7408) specification — AQA (2017)