Standing Wave Calculator
Calculate standing wave frequencies using f_n = nv/(2L) for strings and pipes.
What Is Standing Wave Calculator?
Calculate standing wave frequencies using f_n = nv/(2L) for strings and pipes. is a fundamental physics relationship with wide applications in science and engineering.
The formula arises from first principles and has been verified experimentally across many scales and conditions. Understanding the underlying derivation helps avoid common errors.
Engineers and scientists use this relationship daily in design, analysis, and problem-solving. The calculator handles the arithmetic so you can focus on the physics.
Pay careful attention to units — all formulas require SI units (meters, kilograms, seconds, Newtons, Pascals) unless a specific unit is built into the constant.
Formula Reference Table
| Solve For | Formula | Notes |
|---|---|---|
| Fixed-Fixed (string) | f_n = nv/(2L) | n = 1,2,3... all harmonics |
| Closed-Open (pipe) | f_n = (2n−1)v/(4L) | n = 1,2,3... odd harmonics only |
| Fundamental (n=1) | f₁ = v/(2L) or v/(4L) | Lowest frequency |
| Wavelength in pipe | λ_n = 2L/n (fixed-fixed) | Wavelength of nth mode |
| String tension | v = √(T/μ) | T = tension, μ = mass/length |
| Harmonic ratio | f_n = n·f₁ | nth harmonic is n× fundamental |
3 Worked Examples
String L = 0.65 m, v = 286 m/s. Fundamental and 3rd harmonic.
- f₁ = 286/(2×0.65) = 220 Hz (A3 note!)
- f₃ = 3×220 = 660 Hz
L = 0.39 m, v_air = 343 m/s. Fundamental?
- f₁ = v/(4L) = 343/(4×0.39) = 220 Hz
- Only odd harmonics: f₃ = 660, f₅ = 1100 Hz...
Closed-open pipe, fundamental = A4 = 440 Hz.
- L = v/(4f) = 343/(4×440) = 0.195 m = 19.5 cm
Real-World Applications
Common Mistakes to Avoid
Use SI units throughout.
Check the specific geometry or conditions.
Formula assumes ideal conditions.
Check positive directions.
Keep full precision until final answer.
Frequently Asked Questions
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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.