Standing Wave Calculator
Enter wave speed, length and harmonic number to compute standing-wave frequencies f_n=nv/(2L) and wavelengths λ_n=2L/n. v=340, L=1 → f₁=170 Hz, f₂=340 Hz; guitar v=5000, L=0.65 → f₁≈3846 Hz.
Input Data
Results
At a glance:A standing wave is a non-propagating wave formed by superposing two waves of the same frequency and amplitude travelling in opposite directions. For a string fixed at both ends (or an open-open pipe), the standing-wave frequencies are f_n=n·v/(2L) (n=1,2,3,…), wavelengths λ_n=2L/n, with f_n the n-th harmonic, v the wave speed, L the length and n the harmonic number. n=1 is the fundamental (lowest pitch); n=2,3,… are overtones (harmonics). Physically the wave sets up nodes (zero amplitude) and antinodes (maximum amplitude); for the n-th mode there are n+1 nodes (including both ends) and n antinodes. On a string v=√(T/μ) (T tension, μ linear density). History: Melde demonstrated standing waves with a tuning fork and string in 1860; Helmholtz studied harmonics and timbre. Classic example: v=340 m/s (sound), L=1 m → f₁=170 Hz, f₂=340 Hz, f₃=510 Hz; a guitar string v=5000 m/s, L=0.65 m → f₁=3846 Hz. Applications: (1) string instruments — guitar, violin, piano; (2) wind instruments — open/closed pipe harmonics; (3) microwave resonant cavities — radar, accelerators; (4) quantum mechanics — particle-in-a-box standing waves; (5) room acoustics — resonance modes.
Formula
Standing-wave frequency: f_n = n·v/(2L), n=1,2,3,…
Fundamental: f₁ = v/(2L)
Wavelength: λ_n = 2L/n
String wave speed: v = √(T/μ) (T tension, μ linear density)
Nodes: n+1 (incl. ends); antinodes: n
$$f_n = \frac{n v}{2L}, \quad \lambda_n = \frac{2L}{n}, \quad n = 1,2,3,\ldots$$How to Use
- Enter wave speed v (m/s), length L (m) and harmonic number n.
- The tool gives the fundamental f₁, the n-th harmonic f_n and its wavelength λ_n.
Case Studies
Strings and room acoustics
Guitar string v=5000, L=0.65 → f₁=3846 Hz; pressing a fret shortens L and raises pitch.
Open pipe (flute) L=0.6 → f₁=340/(2×0.6)=283 Hz; closed pipe (clarinet) halves the fundamentals.
A room's standing modes (axial, tangential) cause bass peaks — studio design avoids coincident modes.
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.