Standing Wave Formula
Learn the standing wave formula for strings and open/closed pipes.
Includes frequency equations, harmonics table, and worked examples.
The Formulas
Closed Pipes (odd n only): fₙ = n × (v / 4L)
A standing wave forms when waves reflect back and forth in a confined medium — like a vibrating guitar string or air inside a pipe. The result is a stable pattern of nodes (no movement) and antinodes (maximum movement). Only specific frequencies, called harmonics, can sustain standing waves in a given space.
For strings and open pipes, all harmonics (n = 1, 2, 3, ...) are supported. For closed pipes (open at one end, closed at the other), only odd harmonics (n = 1, 3, 5, ...) are possible.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| fₙ | Frequency of the nth harmonic | Hz |
| n | Harmonic number (1, 2, 3, ... for strings/open pipes; 1, 3, 5, ... for closed pipes) | dimensionless |
| v | Wave speed in the medium | m/s |
| L | Length of the string or pipe | m |
Example 1 — Guitar String (Middle C)
A guitar string is 0.65 m long. The wave speed is 340 m/s. Find the fundamental frequency.
f₁ = 1 × (340 / (2 × 0.65))
f₁ = 340 / 1.30
f₁ ≈ 261.5 Hz — this is middle C!
Example 2 — Closed Pipe (Bass Note)
A closed pipe is 0.85 m long. Speed of sound = 343 m/s. Find the fundamental frequency.
f₁ = 1 × (343 / (4 × 0.85))
f₁ = 343 / 3.40
f₁ ≈ 101 Hz — a deep bass note
Harmonics Table
For a string or open pipe with fundamental frequency f₁:
| n | Harmonic Name | Frequency | Relationship |
|---|---|---|---|
| 1 | Fundamental (1st harmonic) | f₁ | Base frequency |
| 2 | 2nd harmonic (1st overtone) | 2 × f₁ | One octave above |
| 3 | 3rd harmonic (2nd overtone) | 3 × f₁ | Perfect fifth above octave |
| 4 | 4th harmonic (3rd overtone) | 4 × f₁ | Two octaves above |
| 5 | 5th harmonic (4th overtone) | 5 × f₁ | Major third above two octaves |
For a closed pipe, only odd harmonics exist: n = 1, 3, 5, 7, ... This is why clarinets (which behave like closed pipes) produce a distinctly different tone from flutes (open pipes).
The Physics Behind Standing Waves
Standing waves arise from interference between a wave and its reflection. When the length of the medium is exactly right, the reflected wave reinforces the original rather than cancelling it. This creates fixed points of zero displacement (nodes) and maximum displacement (antinodes).
The condition for standing waves on a string: both ends must be nodes (fixed points). This means the string length must equal a whole number of half-wavelengths: L = n × (λ/2).
For a closed pipe, the closed end is always a node and the open end is always an antinode. This different boundary condition is why the closed pipe formula uses 4L instead of 2L, and supports only odd harmonics.
When to Use It
Use the standing wave formula when:
- Designing string instruments — guitars, violins, pianos, and harps all rely on standing waves
- Tuning pipe organs and other wind instruments
- Analyzing room acoustics and identifying resonant frequencies in concert halls
- Designing resonance chambers in industrial and acoustic engineering
- Understanding how microwave ovens, laser cavities, and transmission lines work
- Avoiding unwanted resonances in speaker enclosures or building structures
The standing wave formula is one of the most widely applied equations in all of acoustics and wave physics. From the tuning of a concert grand piano to the design of a recording studio, understanding harmonics and resonance is essential.