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Superposition and Stationary Waves

Pluck a guitar string and, instead of a simple travelling wave, you get a pattern that seems to stand completely still — some points frozen at zero displacement, others oscillating with huge amplitude. This is a stationary wave, and it emerges entirely from two identical progressive waves overlapping and superposing.

40 min Video by ZPhysics Module 4.4 Waves

A Level Physics Waves Revision: Superposition, phase difference, path difference and interferenceWatch the full walkthrough before the notes below.

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What you'll be able to do

State the principle of superposition
Explain how a stationary wave is formed from two progressive waves
Define nodes and antinodes
Use the relationship between the length of a stretched string and the wavelength of its harmonics

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 of each wave. Where two waves are in phase, they add constructively (a larger resultant); where they are in antiphase, they add destructively (a smaller, or zero, resultant).

Stationary waves

A $stationary$ (or standing) $wave$ is formed when two progressive waves of the same frequency and amplitude, travelling in opposite directions, superpose — typically a wave and its own reflection. Unlike a progressive wave, a stationary wave transfers no net energy along its length; the pattern simply oscillates in place.

$Nodes$ are points of permanently zero displacement, where the two component waves are always in antiphase; $antinodes$ are points of maximum oscillation amplitude, where the two component waves are always in phase. Adjacent nodes (or adjacent antinodes) are always separated by exactly half a wavelength.

distance between adjacent nodes = \frac{λ}{2}

Node spacing in a stationary wave.

1Use

node spacing = \frac{λ}{2}

, so

λ = 2 \times node spacing

λ = 2 \times 0.30

Answer

λ = 0.60

Tip — A stationary wave stores energy but does not transport it from place to place — this is exactly what distinguishes it from the progressive waves that combine to form it.

Harmonics on a stretched string

A string fixed at both ends must have a node at each end. The simplest possible stationary wave pattern (the $fundamental$ , or first harmonic) has just one antinode in the middle, so the string length equals half a wavelength. Higher harmonics fit additional half-wavelengths into the same string length.

1At the fundamental, the string length equals half a wavelength:

L = \frac{λ}{2}

2Rearranged:

λ = 2 L = 2 \times 0.80

Answer

λ = 1.60

Equation recap

node spacing = \frac{λ}{2}

Distance between adjacent nodes (or antinodes).

λ = 2 L (fundamental, string fixed at both ends)

Fundamental wavelength on a string.

Common mistakes to avoid

Believing a stationary wave transports energy along its length, like a progressive wave.

A stationary wave has zero net energy transfer — it stores energy, oscillating in place, rather than transporting it.

Confusing nodes and antinodes.

Nodes are points of ZERO displacement (always still); antinodes are points of MAXIMUM displacement (oscillating with the largest amplitude).

Assuming node spacing equals a full wavelength.

Adjacent nodes are separated by HALF a wavelength, not a full wavelength.

Key takeaways

Superposition: resultant displacement is the vector sum of overlapping waves.
A stationary wave forms from two identical progressive waves travelling in opposite directions; it transfers no net energy.
Nodes (zero displacement) and antinodes (maximum displacement) are each separated by λ/2.
A string fixed at both ends has a fundamental wavelength of 2L, where L is the string length.

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Superposition and Stationary Waves

40 min Video by ZPhysics Module 4.4 Waves