Standing Waves & Harmonics
Watch two travelling waves combine into a standing wave, with fixed nodes and oscillating antinodes. Harmonics on strings and in pipes, made visual.
Pluck a guitar string and the wave you send racing along it has nowhere to go — it reflects off the fixed end and runs straight back into itself. The two waves lock together into a shimmering pattern that seems to stand perfectly still. That is a standing wave, and it is why every instrument has a pitch.
When a wave meets its own reflection
Send a wave along a string fixed at both ends and it reflects at each end, so at any moment two identical waves — same frequency, same amplitude — are travelling in opposite directions. By the principle of superposition their displacements add, and the result is a standing (stationary) wave: a fixed pattern that oscillates in place instead of moving along.
Certain points, the nodes, never move at all, because the two waves always cancel there. Midway between them the string swings with maximum amplitude — the antinodes.
A progressive wave walks across the room; a standing wave dances on the spot. Same ingredients, but the reflection pins the pattern in place.
Watch the pattern settle
In the canvas below, switch to the standing-wave view: two waves head in opposite directions and their sum settles into a fixed shape. Watch how the nodes stay glued to their positions while the antinodes swing between the extremes.
Text description ↓Hide text description ↑
A canvas that adds two waves. In the standing-wave mode, two waves travelling in opposite directions superpose to give a fixed pattern of nodes (points that never move) and antinodes (points of maximum oscillation), a quarter of a wavelength apart. The pattern does not travel along the medium.
Harmonics on a string
A string fixed at both ends must have a node at each end, which only lets certain wavelengths fit. The simplest fit — a node at each end and one antinode in the middle — is the fundamental, or first harmonic. Here the string holds exactly half a wavelength, so λ₁ = 2L.
Squeeze in more half-wavelengths and you get the higher harmonics. For a string of length L fixed at both ends the allowed wavelengths and frequencies are:
Worked example
A string of length 1.2 m fixed at both ends has a fundamental frequency of 100 Hz. What is the frequency of the third harmonic?
Higher harmonics are simply whole-number multiples of the fundamental, fₙ = n·f₁. For the third harmonic n = 3:
f₃ = 3 × 100 Hz = 300 Hz.
As a sanity check on the wave itself, the fundamental wavelength is λ₁ = 2L = 2.4 m, so the wave speed is v = f₁λ₁ = 100 × 2.4 = 240 m/s — the same for every harmonic on this string.
Common mistakes
Practice
The fundamental frequency of a pipe is 150 Hz. What is the frequency of its second harmonic?
Show the answer ↓Hide the answer ↑
The second harmonic is twice the fundamental, fₙ = n·f₁ with n = 2: f₂ = 2 × 150 Hz = 300 Hz.
From strings to pipes
The same idea sets the notes of wind instruments, where the standing wave forms in a column of air, and explains resonance in bridges and buildings. Change the length and you change which harmonics fit — exactly what a guitarist does by pressing a fret.
Frequently asked questions
What is a standing wave?+
What is the difference between a node and an antinode?+
What is the fundamental frequency?+
How do standing waves differ from travelling waves?+
The ScholarsGate Physics Team
Oxbridge & Russell Group physics tutors
Written and reviewed by ScholarsGate tutors who teach A-Level and undergraduate physics. Every explainer is checked against the AQA, Edexcel, OCR and CIE specifications.
Keep exploring
Want a tutor to walk you through it?
Book a DBS-checked A-Level Physics tutor for a 1-on-1 lesson — online or in person.
Find a A-Level Physics tutor