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Standing Wave Calculator

Calculate resonance frequencies, wavelengths, nodes, and antinodes for open tubes, closed tubes, and vibrating strings. Visualize the standing wave pattern for any harmonic.

Parameters

m
0.01 m10 m
m/s
n
1st10th
Active Formula
fₙ = n × v / (2L)
Open tube: all harmonics present

Results

Selected Harmonic Frequency
171.5
Hz
Fundamental f₁
171.5 Hz
Wavelength λ
2.00 m
Nodes
2
Antinodes
1

Standing Wave Pattern

N = Node (zero displacement) A = Antinode (maximum displacement)

First 8 Harmonics

Harmonic (n) Frequency (Hz) Wavelength (m) Nodes Antinodes Name

Understanding Standing Waves

A standing wave forms when two waves of the same frequency travel in opposite directions and interfere. Unlike a traveling wave, the pattern appears stationary — certain points (nodes) never move while others (antinodes) oscillate with maximum amplitude.

In a tube or on a string, standing waves only form at frequencies where the boundary conditions are satisfied. For an open tube, both ends must be antinodes (pressure nodes). For a closed-end tube, the open end is an antinode and the closed end a node — this eliminates even harmonics and gives a distinctive hollow timbre. Strings fixed at both ends require nodes at both endpoints.

Musical Instrument Applications

  • Organ Pipes — Open pipes support all harmonics (f, 2f, 3f…), giving a bright tone. Stopped (closed) pipes support only odd harmonics (f, 3f, 5f…), producing a rounder, hollow sound one octave lower for the same pipe length.
  • Guitar & String Instruments — A plucked string vibrates at its fundamental plus harmonics determined by the string length, tension, and linear density. The fret positions correspond to exact fractional string lengths (1/2 for the octave, 2/3 for the fifth, etc.).
  • Wind Instruments — Clarinets behave as closed-end tubes (odd harmonics only), while flutes and trumpets behave as open tubes. This accounts for their different timbres despite similar playing ranges.
  • Room Acoustics — Room modes are standing waves between parallel walls. Low frequencies that fit the room dimensions as half-wavelengths (λ/2) are reinforced, causing uneven bass response at different listening positions.

Frequently Asked Questions

What is the difference between open and closed tube resonances?
An open tube (open at both ends) supports all integer harmonics: f₁, 2f₁, 3f₁, etc. The fundamental frequency is f₁ = v/(2L). A tube closed at one end only supports odd harmonics: f₁, 3f₁, 5f₁, etc. Its fundamental is f₁ = v/(4L) — half the frequency of an equal-length open tube, so it sounds an octave lower. Closed tubes are used in stopped organ pipes and clarinets.
How do I find the fundamental frequency of a vibrating string?
For a string fixed at both ends: f₁ = v/(2L), where v = √(T/μ) with T = string tension (N) and μ = linear mass density (kg/m). This calculator assumes you know the wave speed directly. For a guitar string, higher tension or lower mass density raises the pitch. Doubling the tension raises the pitch by a factor of √2 (about 6 semitones).
What are nodes and antinodes in a standing wave?
Nodes are points of zero displacement where the two interfering waves cancel completely — they never move. Antinodes are points of maximum displacement where the waves reinforce — they oscillate between +A and −A. In a pressure wave (tube), nodes are pressure antinodes and vice versa. The number of antinodes equals the harmonic number n for open tubes and strings, but equals (2n−1)/2 for closed tubes (fractional antinodes at walls).
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