∿ Standing waves
String vibration modes: nodes and antinodes, the traveling waves that make up a standing wave, the harmonic series and a calculator for real strings.
Mode n3
Amplitude0.60
Damping0.020
Animation speed×1.0
View:
Ends:
Animation frequency:
Mode:
Examples:
Parameters of the selected mode
—
Mode n
—
Nodes (incl. ends)
—
Antinodes
—
Wavelength λn (L — string length)
—
Frequency fn
—
Vibration regime
Fixed ends: λn = 2L / n, fn = n · v / 2L = n · f₁
One end free: λn = 4L / (2n − 1), fn = (2n − 1) · v / 4L
Wave speed on a string: v = √(T / μ)
One end free: λn = 4L / (2n − 1), fn = (2n − 1) · v / 4L
Wave speed on a string: v = √(T / μ)
- A standing wave appears when a wave traveling along the string is reflected from the end and superposes with the incident wave. In the “Traveling components” view you can see both waves (the purple one moves right, the orange one left), while their sum, the blue curve, does not move along the string.
- Yellow dots are nodes, where the string does not move. Between the nodes are antinodes with the largest swing. A fixed end is always a node, a free end is always an antinode.
- In the “Free” mode (as in the original) a plucked string decays as e−γt: at γ = 0.02 the amplitude falls by a factor of e in about 50 s. Changing the mode excites the string anew. In the “Driven” mode the amplitude is steady, as if an external force at the resonant frequency sustained the mode.
- The “All harmonics” view shows the first eight modes at once. By default all of them oscillate at the same rate, as in the original (cos 2.2·t, period 2.86 s). The “Proportional to n” mode shows the physical dependence fn ∝ n: each next mode oscillates faster and has one more node.
- The “Sound” button plays a tone at the frequency fn for f₁ = 110 Hz (the note A): for n = 4 this is 440 Hz.
Enter the parameters of a real string: length, tension and linear density. You will get the wave speed, the fundamental frequency, the harmonic frequencies and the positions of the nodes.
String length L, m
Tension T, N
Linear density μ, g/m
Harmonic number n
String ends
Examples:
—
Wave speed v, m/s
—
Fundamental frequency f₁, Hz
—
Frequency fn, Hz
—
Wavelength λn, m
—
Nodes / antinodes
—
Nodes from the fixed end, m
v = √(T / μ), f₁ = v / 2L (fixed ends) or v / 4L (one free)
fn = n · f₁ (fixed) or (2n − 1) · f₁ (one free)
fn = n · f₁ (fixed) or (2n − 1) · f₁ (one free)
Harmonic series: fixed ends and one free end
| n | Fixed: f / f₁ | Nodes | Antinodes | One free: f / f₁ | Nodes | Antinodes |
|---|---|---|---|---|---|---|
| 1 | 1 | 2 | 1 | 1 | 1 | 1 |
| 2 | 2 | 3 | 2 | 3 | 2 | 2 |
| 3 | 3 | 4 | 3 | 5 | 3 | 3 |
| 4 | 4 | 5 | 4 | 7 | 4 | 4 |
| 5 | 5 | 6 | 5 | 9 | 5 | 5 |
| 6 | 6 | 7 | 6 | 11 | 6 | 6 |
Standing Waves on a String Online: Simulation and Harmonics Calculator
A free interactive tool that shows how standing waves form on a string. Choose the mode number, amplitude and damping, look at the nodes and antinodes, decompose a standing wave into two traveling ones and compare the first eight harmonics. You can fix both ends or leave one free, turn on the sound, and calculate the fundamental frequency and harmonics for a real guitar string, violin string or any other string.
Key concepts
Nodes and antinodes
At nodes the string is at rest; at antinodes it vibrates with the largest amplitude. The distance between adjacent nodes equals half a wavelength.
n = 4 → 5 nodes, 4 antinodes
Harmonics
The modes of a string with fixed ends have frequencies that are multiples of the fundamental. It is the set of harmonics that determines the timbre of an instrument.
fn = n · f₁
Wave speed on a string
It depends on the tension and linear density: the tighter and lighter the string, the faster the wave and the higher the pitch.
v = √(T / μ)
Where it is used
- String instruments: guitar, violin, piano, harp
- Wind instruments and organ pipes (standing pressure waves in an air column)
- Laser and microwave oven resonators
- Tuning instruments and calculating strings by tension
- Quantum mechanics: stationary states of a particle in a potential well are analogous to string modes
- Laboratory work in acoustics and wave physics
How to use
- Move the “Mode n” slider and count the nodes and antinodes
- Switch the view to “Traveling components” to see which waves make up the standing wave
- Turn on “All harmonics” and compare the frequencies of different modes
- Choose “Right end free” and watch how the harmonic series changes
- Press “Sound” to hear the tone of the selected mode
- In the “Calculator” tab enter the length, tension and density of your string
Frequently asked questions
What is a standing wave?
It is an oscillation in which the wave pattern does not travel along the medium but stays in place: at the stationary nodes the amplitude is zero, and between them the antinodes oscillate with the maximum swing. A standing wave arises as the sum of two identical waves traveling toward each other.
Why do only certain frequencies occur on a string?
At fixed ends the string cannot move, so a whole number of half-waves must fit along the length L: L = n·λ/2. Hence the wavelengths λₙ = 2L/n and the frequencies fₙ = n·v/(2L). Other frequencies do not give a stable pattern and die out quickly.
How does a node differ from an antinode?
At a node the displacement of the string is always zero, at an antinode it is maximal. The distance between adjacent nodes and between adjacent antinodes equals λ/2. A fixed end is always a node, and a free end is always an antinode.
How do you calculate the fundamental frequency of a string?
First find the wave speed v = √(T/μ), where T is the tension in newtons and μ is the linear density in kg/m. For a string with two fixed ends f₁ = v/(2L). So a shorter, tighter and lighter string sounds higher. The calculator on this page does it automatically.
Does a standing wave carry energy?
On average, no: the incident and reflected waves carry equal energy fluxes in opposite directions, so there is no net flow. Energy shifts between kinetic (near the antinodes) and potential (deformation near the nodes), but does not flow along the string.