Standing Wave Resonance Calculator

Calculate harmonic frequencies, wavelengths, and resonant modes for strings and open/closed pipes.

Sound in air=343, String: √(T/μ)
Fundamental: n=1
Please check your inputs and try again.

Boundary Conditions Select Discrete Standing-Wave Frequencies

Standing waves form when waves of the same frequency travel in opposite directions and interfere, creating fixed nodes and antinodes. On a string fixed at both ends, allowed wavelengths satisfy L=nλ/2, giving fn=nv/(2L). An open-open pipe has the same harmonic pattern for displacement, while a pipe closed at one end supports only odd harmonics with fn=nv/(4L) for n=1,3,5,…

Wave speed comes from the medium: v=√(T/μ) on an ideal stretched string and approximately the speed of sound in an air column. Resonance occurs when a driving frequency matches an allowed natural mode.

String/open-open: fn=nv/(2L);   closed-open: fn=nv/(4L), n odd
SymbolMeaningWhy it appears / units
LResonator lengthm.
vWave speedm/s.
nMode or harmonic indexPositive integer; odd-only for ideal closed-open pipe.
fnResonant frequencyHz.

Higher modes contain more nodes and shorter wavelengths. Real pipes need end corrections, and real strings have stiffness and loss, so measured frequencies can differ slightly from the ideal formulas.

The fundamental wavelength should fit the boundary geometry. Fixed-fixed or open-open systems contain half a wavelength in the fundamental length L, while a closed-open tube contains one quarter wavelength. Drawing nodes and antinodes before calculating is an effective check against a factor-of-two error.

Worked Examples

Example 1: Guitar string: L=0.65m, v=343m/s... (use string wave speed √(T/μ))
v≈400m/s (typical E string), n=1
Result: f₁=400/(2×0.65)=308 Hz, ≈E4
Fundamental of guitar string
Example 2: Closed organ pipe L=0.5m, n=2 (3rd harmonic)
f=(2×2-1)×343/(4×0.5)
Result: f=514.5 Hz — 3rd harmonic only
Closed pipes lack even harmonics
Example 3: Fixed string fundamental
L=0.65m, v=260m/s
Result: f1=200Hz
The fundamental has half a wavelength along the string length.
Example 4: Closed pipe first overtone
L=0.50m, v=340m/s, n=3
Result: f3=510Hz
The next allowed mode is the third harmonic, not the second.

Common Mistakes

⚠️
Using every integer harmonic for a closed-open pipe

The ideal closed-open boundary conditions support n=1,3,5,… only.

⚠️
Confusing nodes and antinodes between pressure and displacement in air columns

At an open end, displacement is an antinode while pressure variation is a node; the pattern reverses at a closed end.

⚠️
Using frequency formula without the correct wave speed

String wave speed depends on tension and linear density, while air-column speed depends on sound speed and temperature.

Frequently Asked Questions

Why no even harmonics in closed pipe?
Closed end must be a displacement node; open end must be an antinode. Only odd multiples of λ/4 fit: λ/4, 3λ/4, 5λ/4... giving only odd harmonics. This gives closed pipes a distinctive 'hollow' timbre.
String wave speed?
v=√(T/μ) where T=tension (N) and μ=mass per unit length (kg/m). Guitar tuning changes T → changes v → changes frequency. Heavier strings (higher μ) have lower frequency at same tension.
Why does a string fixed at both ends have nodes at the ends?
The endpoints cannot move, so displacement must be zero there. Only wavelengths that fit an integer number of half-wavelengths between those nodes are allowed.
Why are even harmonics absent in an ideal closed pipe?
One end must be a displacement node and the other an antinode, which fits odd numbers of quarter-wavelengths rather than all half-wavelength multiples.
What is the difference between resonance and a standing wave?
A standing wave is the spatial interference pattern. Resonance is the strong response that occurs when a driving frequency matches an allowed natural frequency.
Why do real pipe frequencies need end correction?
The oscillating air extends slightly beyond an open physical end, making the acoustic effective length longer than the measured tube length.
How do boundary conditions change standing-wave resonances?
A string fixed at both ends and an open-open air column support integer harmonics fn=nv/(2L). A pipe closed at one end supports only odd harmonics in the ideal model, fn=nv/(4L) for n=1,3,5,… . Using the wrong boundary condition changes every resonant frequency.

Formula Explorer connections

Interpretation: This relationship connects frequency, wavelength, speed, phase, intensity or resonance in an oscillating system. Assumption: Identify the medium, boundary conditions and reference frame. Linear waves, small amplitudes, nondispersive media or ideal resonance may be assumed.

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