Enter values to see results
—
Hz
—
m
—
Hz
—
—
—
m
Enter values to see results
—
Hz
—
m
—
Hz
—
—
—
m
The Standing Wave Calculator computes the resonant frequencies and wavelengths of standing waves on strings and in pipes. Standing waves form when two waves of equal amplitude and frequency traveling in opposite directions interfere, creating fixed nodes (zero displacement) and antinodes (maximum displacement). The resonant frequencies for a string or pipe of length $$L$$ are:
$$f_n = \frac{nv}{2L} \quad \text{(both ends fixed or both open)}$$
$$f_n = \frac{(2n-1)v}{4L} \quad \text{(one end open, one closed)}$$
Standing waves are the physical basis of musical instruments. A guitar string vibrates at frequencies determined by its length, tension, and mass — each harmonic $$n$$ corresponds to a richer pattern of vibration. Wind instruments use air columns in pipes, where the boundary conditions (open vs. closed ends) determine which harmonics are present.
Beyond music, standing waves appear in microwave ovens (the hot spots are antinodes), laser cavities (resonant modes between mirrors), quantum mechanics (electron standing waves in atoms), and earthquake engineering (resonant frequencies of buildings). This calculator supports three boundary conditions: both ends fixed (strings, closed pipes), one end open (clarinet-type), and both ends open (flute-type).
The standing wave patterns depend on boundary conditions:
Both ends fixed (string, closed pipe): Both endpoints are nodes. The wavelength of the $$n$$-th harmonic is $$\lambda_n = 2L/n$$ and the frequency is:
$$f_n = \frac{nv}{2L}, \quad n = 1, 2, 3, \ldots$$
All integer harmonics are present.
One end open, one closed (clarinet-type pipe): The closed end is a node; the open end is an antinode. Only odd harmonics exist:
$$f_n = \frac{(2n-1)v}{4L}, \quad n = 1, 2, 3, \ldots$$
This gives the series: $$f_1, 3f_1, 5f_1, 7f_1, \ldots$$
Both ends open (flute-type pipe): Both endpoints are antinodes. Like the fixed-fixed case, all harmonics are present:
$$f_n = \frac{nv}{2L}, \quad n = 1, 2, 3, \ldots$$
The fundamental frequency ($$n=1$$) is the lowest resonant frequency and determines the perceived pitch. Higher harmonics (overtones) add timbre — the characteristic tone quality that distinguishes a trumpet from a violin playing the same note.
The number of nodes and antinodes increases with harmonic number. For fixed-fixed strings, the $$n$$-th harmonic has $$n+1$$ nodes and $$n$$ antinodes. The node spacing equals half the wavelength: $$\lambda_n / 2 = L/n$$. For an open-closed pipe, the fundamental wavelength is $$4L$$ (four times the pipe length), which is why a closed pipe sounds an octave lower than an open pipe of the same length.
Inputs
Results
A guitar A-string (65 cm, v = 286 m/s) has a fundamental of 220 Hz (A3). The string vibrates in one arc with nodes at the nut and bridge.
Inputs
Results
A 60 cm clarinet-type pipe at the 3rd harmonic (n=3, which is the 5th partial) produces about 715 Hz. Only odd harmonics resonate in an open-closed pipe, giving the clarinet its distinctive hollow tone.
A closed end must be a node (zero displacement) and an open end must be an antinode (maximum displacement). The shortest wave fitting these constraints has $$\lambda = 4L$$ (quarter-wavelength fits in the pipe). The next allowed mode has $$\lambda = 4L/3$$, then $$4L/5$$, etc. These correspond to the 1st, 3rd, 5th harmonics — all odd. Even harmonics require both ends to be the same type (both nodes or both antinodes), which contradicts the mixed boundary condition.
The fundamental is the 1st harmonic. The 1st overtone is the 2nd harmonic, the 2nd overtone is the 3rd harmonic, and so on. For open-closed pipes, the 1st overtone is the 3rd harmonic (since even harmonics are absent). So overtone numbering depends on boundary conditions, while harmonic numbering always refers to integer multiples of the fundamental.
Halving the string length doubles the fundamental frequency (raises pitch by one octave): $$f_1 = v/(2L)$$. This is the principle behind fretting on a guitar — pressing a string against a fret shortens the vibrating length, raising the pitch. The 12th fret is at the midpoint, producing a note exactly one octave higher.
Nodes are points of zero displacement where destructive interference permanently cancels the wave. Antinodes are points of maximum displacement where constructive interference produces the largest oscillation. In a standing wave, nodes and antinodes alternate at regular intervals of $$\lambda/4$$.
When a string or air column is excited (plucked, bowed, blown), it vibrates simultaneously in multiple modes — the fundamental plus higher harmonics. The relative amplitude of each harmonic depends on how and where the excitation occurs. This mixture of harmonics produces the instrument's unique timbre, which is why a piano and a flute sound different even playing the same note.
Standing waves extend to higher dimensions as Chladni patterns (on vibrating plates), drum head modes (circular membranes), and room acoustics (3D cavities). In 2D, the nodal lines form complex patterns visible by sprinkling sand on a vibrating plate. In 3D, microwave ovens have standing wave patterns that create hot and cold spots.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
How helpful was this calculator?
Be the first to rate!
