Standing Waves & Resonance Calculator
Calculate resonant frequencies, harmonics, and wavelengths for standing waves in strings, open pipes, and closed pipes.
Standing Waves Guide
Standing Wave Conditions
Standing waves form when two identical waves travel in opposite directions and interfere. Nodes: points of zero displacement (destructive interference). Antinodes: points of maximum displacement (constructive interference). For a string fixed at both ends (or open pipe): both ends are nodes (or antinodes). Resonant lengths: L = nλ/2 for n = 1, 2, 3... Fundamental (n=1): L = λ/2. Second harmonic: L = λ. nth harmonic: L = nλ/2. Fundamental frequency: f₁ = v/(2L). All harmonics present: f_n = nf₁.
Closed Pipes
A closed pipe has one closed end (node) and one open end (antinode). Only odd harmonics are present. Resonant condition: L = (2n-1)λ/4 for n = 1, 3, 5... Fundamental: L = λ/4. 3rd harmonic: L = 3λ/4. Fundamental frequency: f₁ = v/(4L). Harmonics: f₁, 3f₁, 5f₁, 7f₁... Example: clarinet behaves as a closed pipe — lacks even harmonics — producing its characteristic hollow tone. A flute (open both ends) has all harmonics — brighter, more complex sound. Organ pipes: open pipes produce full harmonic s
Resonance in Musical Instruments
Guitar string (L=0.65m, v=400 m/s): f₁ = 400/(2×0.65) = 307.7 Hz (approximately D4). To get A4 (440Hz): L = 400/(2×440) = 0.454m (shorten by pressing fret). Speed of waves in a string: v = √(T/μ). T = tension (N), μ = linear mass density (kg/m). Tightening a string increases tension → increases wave speed → increases frequency. Thicker strings (higher μ) have slower waves → lower frequency → lower pitch. The harmonic series explains why musical intervals sound consonant: frequency ratios 1:2 (oc
End Correction for Pipes
Real pipes have an end correction: the effective length is slightly longer than the physical pipe due to air motion beyond the open end. End correction: Δ ≈ 0.6r (where r = pipe radius). Effective length L_eff = L + 2Δ (both ends open) or L + Δ (one end). This correction is small but measurable — important in precision wind instrument design. Resonance demonstrates: standing waves arise when the round-trip distance equals a whole number of wavelengths. Natural frequency: all physical systems hav
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