Standing (Stationary) Waves

Pluck a guitar string and it sings a clear, steady note. Look closely and something strange is going on: the string is bulging up and down, yet the pattern of bulges does not race along the string the way a ripple races across a pond. The humps sit in the same places — some points on the string never move at all, while others thrash up and down as hard as they can. The wave seems to be standing still.

This is a standing wave (also called a stationary wave). It is one of the most important patterns in physics: it is how every stringed and wind instrument chooses its notes, how a microwave oven has hot and cold spots, and how an opera singer can shatter a wine glass with a single held note. On this page we'll build it from something you already know — superposition — and turn it into exact formulae for the frequencies a string can play.

How a standing wave is made

A standing wave forms when two progressive waves of the same frequency and amplitude travel in opposite directions through the same medium and superpose. In practice you rarely need two separate sources: send one wave down a string (or a sound wave down a pipe) and let it reflect off the far end. The reflected wave travels back through the outgoing wave, and where they overlap they add together point by point.

Write the outgoing wave and its reflection as

y_1 = A\sin(kx - \omega t), \qquad y_2 = A\sin(kx + \omega t).

Superposition says the string's shape is the sum y = y_1 + y_2. Using the identity \sin P + \sin Q = 2\sin\!\big(\tfrac{P+Q}{2}\big)\cos\!\big(\tfrac{P-Q}{2}\big),

y = 2A\sin(kx)\cos(\omega t).

Read this carefully — it is the whole idea in one line. The x and the t have separated. There is no travelling (kx - \omega t) bundle any more. Instead:

The factor 2A\sin(kx) is a fixed shape along the string — it tells you the amplitude at each position x, and it never moves.
The factor \cos(\omega t) just makes that whole shape swell and shrink in step, so every point oscillates with the same frequency but keeps its own place.

Nodes and antinodes

Because the amplitude is set by 2A\sin(kx), some points are special.

A node is a point that is always at zero displacement. It happens wherever \sin(kx) = 0, i.e. kx = 0, \pi, 2\pi, \dots The string there is pinned still no matter what t does.
An antinode is a point of maximum amplitude 2A. It happens wherever \sin(kx) = \pm 1, i.e. kx = \tfrac{\pi}{2}, \tfrac{3\pi}{2}, \dots These points swing through the largest arc.

Since k = \tfrac{2\pi}{\lambda}, consecutive zeros of \sin(kx) are a distance \tfrac{\pi}{k} = \tfrac{\lambda}{2} apart. So the distance between two adjacent nodes is \tfrac{\lambda}{2} — and likewise between two adjacent antinodes. A node and its neighbouring antinode are \tfrac{\lambda}{4} apart. This is the most useful measuring rule on the whole page: count half-wavelengths between the nodes.

Three traps snare almost everyone the first time they meet standing waves:

A standing wave transfers no net energy along the string. A progressive wave carries energy from one place to another; a standing wave just stores it, sloshing between kinetic and potential as the string swells and flattens. Nothing is delivered down the line — the nodes never move, so no energy can flow past them.
The gap between adjacent nodes is \tfrac{\lambda}{2}, not \lambda. One full wavelength spans two loops (node → antinode → node → antinode → node). Reading one loop as a whole wavelength is the single most common mistake in exam calculations — it doubles your wavelength and halves your frequency.
Every point between two neighbouring nodes moves in phase. They all reach the top together and pass through the middle together — they only differ in how far they swing. Nothing is "travelling"; the crest does not slide sideways. (Cross a node and the motion flips to antiphase.)

Progressive versus standing waves

It is worth laying the two kinds of wave side by side, because they are easy to muddle and an exam loves to test the difference.

Energy: a progressive wave transports energy along its direction of travel; a standing wave transports none — energy stays put, trapped between the nodes.
Amplitude: in a progressive wave every point oscillates with the same amplitude; in a standing wave the amplitude varies with position, from 0 at a node to 2A at an antinode.
Phase: in a progressive wave neighbouring points are slightly out of step (the phase lags smoothly along it); in a standing wave a whole loop moves in phase, and adjacent loops move in antiphase.
Waveform: a progressive wave's profile moves; a standing wave's profile stays fixed and merely changes height.
Frequency: both oscillate at the source frequency, but a standing wave can only exist at certain frequencies set by the boundaries — which is exactly what we work out next.

A string fixed at both ends: the harmonics

Clamp a string of length L at both ends — a guitar, violin or piano string. The ends cannot move, so each end must be a node. That single condition is a huge restriction: only standing waves that fit a whole number of loops between the two fixed ends are allowed. Each allowed pattern is a harmonic.

The fundamental — the first harmonic — is the simplest fit: a single loop, one antinode in the middle, a node at each end. Half a wavelength spans the whole string, so

L = \frac{\lambda_1}{2} \quad\Longrightarrow\quad \lambda_1 = 2L.

Using the wave equation v = f\lambda (with v the wave speed on the string), the fundamental frequency is

f_1 = \frac{v}{\lambda_1} = \frac{v}{2L}.

Fit two loops in and you have the second harmonic, with L = \lambda_2 so \lambda_2 = L and f_2 = \tfrac{v}{L} = 2f_1. In general the nth harmonic packs n loops into the string:

L = n\cdot\frac{\lambda_n}{2} \quad\Longrightarrow\quad \lambda_n = \frac{2L}{n}, \qquad f_n = \frac{v}{\lambda_n} = \frac{nv}{2L} = n f_1.

So a fixed string plays a whole-number ladder of frequencies f_1, 2f_1, 3f_1, \dots — and nothing in between. That evenly-spaced ladder is what makes a plucked string sound musical rather than like noise.

A string of length L fixed at both ends (a node at each end), carrying waves of speed v, supports standing waves only at the discrete harmonic frequencies:

Wavelengths: \lambda_n = \dfrac{2L}{n} for n = 1, 2, 3, \dots
Frequencies: f_n = \dfrac{nv}{2L} = n f_1, an integer multiple of the fundamental f_1 = \dfrac{v}{2L}.
Structure: the nth harmonic has n antinodes and n+1 nodes, adjacent nodes being \tfrac{\lambda_n}{2} apart.

See the harmonics for yourself

Here is a string of length L = 1\,\text{m} carrying waves at speed v = 200\,\text{m s}^{-1}, so the fundamental is f_1 = \tfrac{200}{2\times 1} = 100\,\text{Hz}. Switch between the first, second and third harmonics and watch the pattern rebuild: the ends stay pinned as nodes, the number of loops climbs, and the frequency jumps in equal 100\,\text{Hz} steps because f_n = n f_1. The faint mirror curve shows how far each point swings.

Worked examples

Example 1 — fundamental frequency. A guitar string of length L = 0.65\,\text{m} carries transverse waves at v = 520\,\text{m s}^{-1}. Find the fundamental frequency.

f_1 = \frac{v}{2L} = \frac{520}{2 \times 0.65} = \frac{520}{1.30} = 400\,\text{Hz}.

Example 2 — a higher harmonic. What is the frequency of the third harmonic of that same string, and how many nodes does it have?

f_3 = 3 f_1 = 3 \times 400 = 1200\,\text{Hz}, \qquad \text{nodes} = n + 1 = 4.

Example 3 — find the wave speed. A string of length L = 1.2\,\text{m} is seen vibrating in its fourth harmonic at f_4 = 300\,\text{Hz}. Find the wavelength and the wave speed.

\lambda_4 = \frac{2L}{n} = \frac{2 \times 1.2}{4} = 0.60\,\text{m}, \qquad v = f_4\,\lambda_4 = 300 \times 0.60 = 180\,\text{m s}^{-1}.

Example 4 — which harmonic? A string has a fundamental of f_1 = 110\,\text{Hz}. It is driven at 550\,\text{Hz} and a clean standing wave appears. Which harmonic is it?

n = \frac{f_n}{f_1} = \frac{550}{110} = 5 \quad\Rightarrow\quad \text{the 5th harmonic.}

Pipes: air columns instead of strings

The same physics runs a flute, an organ pipe or a bottle you blow across — only now the standing wave is a sound wave in a column of air, and the rule for the ends changes:

A closed end (a sealed end) forces the air to be still, so it is a displacement node.
An open end lets the air move freely, so it is a displacement antinode.

A pipe open at both ends (antinode–antinode) mirrors the fixed string: it fits \lambda_n = \tfrac{2L}{n} and plays the full ladder f_n = \tfrac{nv}{2L}. A pipe closed at one end (node–antinode) is the odd one out: its fundamental fits only a quarter-wavelength, \lambda_1 = 4L, so f_1 = \tfrac{v}{4L}, and only the odd harmonics survive, f_n = \tfrac{n v}{4L} for n = 1, 3, 5, \dots That missing set of even harmonics is why a clarinet (closed–open) sounds hollow and woody next to a flute (open–open).

When you bow or pluck a string you don't feed it a pure single frequency; you jolt it with a mess of them. The string simply rejects every frequency that doesn't fit between its fixed ends, and resonates at the ones that do — f_1, 2f_1, 3f_1, \dots The lowest, the fundamental, is the pitch you hear; the higher harmonics blend in to give the instrument its timbre, the reason a violin and a guitar playing the same note still sound utterly different. Shorten the string with your finger and you raise f_1 = \tfrac{v}{2L} by cutting L; tighten it with the tuning peg and you raise v — both push the whole ladder up.

A wine glass plays the same trick in three dimensions. Its rim has one natural frequency at which a standing wave of the glass wall will happily grow. Sing that exact note loudly enough and you keep pumping energy in on every cycle — resonance drives the antinodes to swing wider and wider until the glass flexes past what it can survive and shatters. Same maths as the string; a far more dramatic antinode.