3.3.1.3WavesCore

Superposition and Stationary Waves

When two waves meet, their displacements simply add — the $principle of superposition$ . If a wave meets its own reflection, this adding-up locks the energy in place and forms a $stationary wave$ : a pattern of points that never move (nodes) and points that swing with maximum amplitude (antinodes). It is the physics behind every stringed instrument and organ pipe.

40 min Video by Science Shorts 3.3.1 Progressive and stationary waves

Stationary Waves & Phase — A-Level PhysicsWatch the full walkthrough before the notes below.

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What you'll be able to do

State the principle of superposition of waves
Explain how a stationary wave forms from two identical counter-propagating waves
Identify nodes and antinodes and relate their spacing to wavelength
Contrast stationary and progressive waves
Find the harmonics of a string and use the first-harmonic frequency formula

The principle of superposition

The $principle of superposition$ states that when two or more waves meet at a point, the resultant displacement is the $vector sum$ of the individual displacements.

Where two crests (or two troughs) coincide, the displacements add to give a larger amplitude — $constructive$ superposition. Where a crest meets a trough of equal size, they cancel — $destructive$ superposition. After passing through each other the waves carry on completely unchanged.

y = y_{1} + y_{2}

Resultant displacement is the sum of the individual displacements.

Tip — Superposition is momentary — it describes what happens at the instant the waves overlap. The waves themselves are not permanently changed.

How a stationary wave forms

A $stationary (standing) wave$ forms when two progressive waves of the $same frequency and amplitude$ travel in $opposite directions$ and superpose. In practice this happens when a wave reflects back on itself — for example on a string fixed at both ends, or in a tube.

Unlike a progressive wave, a stationary wave does $not$ transfer energy along its length. The energy stays trapped, sloshing between the kinetic and potential stores of the oscillating medium.

Tip — Two conditions to quote for a stationary wave: two waves of equal frequency and amplitude, travelling in opposite directions.

Nodes and antinodes

A stationary wave has fixed points called $nodes$ where the two waves always cancel, so the displacement is permanently zero. Halfway between them are $antinodes$ , where the waves always reinforce and the amplitude is greatest.

The distance between two adjacent nodes (or two adjacent antinodes) is $half a wavelength$ . Every point between two nodes oscillates in phase with each other, and in antiphase with the points in the next loop.

node-to-node distance = \frac{λ}{2}

Adjacent nodes (or antinodes) are half a wavelength apart.

1Adjacent nodes are separated by

λ /2

λ /2 = 0.30

λ = 0.60

Answer

λ = 0.60

Harmonics on a string

A string fixed at both ends must have a node at each end. The simplest pattern that fits is a single loop — the $first harmonic$ (fundamental) — with wavelength $2 L$ . Higher harmonics fit two loops, three loops, and so on, at whole-number multiples of the first-harmonic frequency.

The first-harmonic frequency rises if you shorten the string, increase its tension $T$ , or use a lighter string (smaller mass per unit length $μ$ ).

f_{1} = \frac{1}{2 L} \frac{T}{μ}

First-harmonic frequency of a string (μ = mass per unit length).

1Use

f_{1} = \frac{1}{2 L} \frac{T}{μ}

T / μ = 80/ (1.2 \times 1 0^{- 3}) = 66700 \approx 258

m s⁻¹.

f_{1} = \frac{258}{2 \times 0.65} = \frac{258}{1.30} \approx 199

Hz.

Answer

f_{1} \approx 200

Tip — The $n$ th harmonic has frequency $f_{n} = n f_{1}$ and fits $n$ loops (antinodes) between the fixed ends.

Equation recap

y = y_{1} + y_{2}

Superposition: displacements add.

node spacing = \frac{λ}{2}

Adjacent nodes are half a wavelength apart.

f_{1} = \frac{1}{2 L} \frac{T}{μ}

First-harmonic frequency of a fixed string.

f_{n} = n f_{1}

Harmonic frequencies are whole-number multiples of f₁.

Common mistakes to avoid

Thinking a stationary wave transfers energy along its length.

A stationary wave stores energy in place — no net energy is transferred along it, unlike a progressive wave.

Taking the node-to-node distance as a full wavelength.

Adjacent nodes (or antinodes) are only λ/2 apart. A full wavelength spans two node-to-node gaps.

Assuming any two waves make a stationary wave.

They must have the same frequency and amplitude and travel in opposite directions (usually a wave and its reflection).

Key takeaways

Superposition: when waves meet, displacements add (vector sum).
A stationary wave forms from two identical waves travelling in opposite directions.
It stores energy — nodes stay still, antinodes oscillate with maximum amplitude, and adjacent nodes are λ/2 apart.
A fixed string resonates at f₁ = (1/2L)√(T/μ) and its whole-number harmonics.

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Superposition and Stationary Waves

When two waves meet, their displacements simply add — the

principle of superposition

. If a wave meets its own reflection, this adding-up locks the energy in place and forms a

stationary wave

: a pattern of points that never move (nodes) and points that swing with maximum amplitude (antinodes). It is the physics behind every stringed instrument and organ pipe.

40 min Video by Science Shorts 3.3.1 Progressive and stationary waves