Standing Waves: Harmonics on a Fixed String
1 · Predict
A string fixed at both ends resonates at specific frequencies called harmonics. How does the frequency of the 2nd and 3rd harmonics relate to the fundamental?
- Each harmonic's frequency is a whole-number multiple of the fundamental.
- Harmonic frequencies follow no simple pattern relative to the fundamental.
- Each harmonic's frequency is half the previous one.
2 · Set Up
- Open the standing-string preset and press Reset. Tension (120 N), linear density (0.01 kg/m), and length (1.2 m) are fixed.
- Enable the fundamental-frequency readout on the string.
- Set the harmonic number n for each trial and record the resonant frequency.
3 · Collect Data
| Harmonic number n | Fundamental f₁ (Hz) | Resonant frequency fₙ (Hz) |
|---|---|---|
| 1 | ||
| 2 | ||
| 3 |
Plot resonant frequency fₙ (y-axis) against harmonic number n (x-axis) for your three trials. Does the line pass through the origin?
4 · Analyze
- Compute f₁ = √(T/μ)/(2L) using T = 120 N, μ = 0.01 kg/m, L = 1.2 m, then fₙ = n·f₁ for each row. Compare to the table.
- Your fₙ-vs-n graph should be a straight line through the origin with slope f₁. Explain why fixing both ends forces only whole-number multiples of f₁ to resonate.
5 · Extend
- A guitar string's fundamental sets the note you hear, but the string also vibrates at higher harmonics simultaneously, giving the note its timbre. Where else in this app can you see multiple harmonics combined?
- If you shortened the string to half its length (keeping tension and linear density the same), what would happen to the fundamental frequency? Use f₁ = v/(2L) to explain.
The Physics Behind This Experiment
Fixed-End Standing-Wave Fundamental
A string fixed at both ends supports a standing wave whose lowest (fundamental) frequency depends on wave speed and string length: f₁ = v/(2L).
Harmonic Series
Higher resonant modes occur at whole-number multiples of the fundamental: fₙ = n·f₁. Each integer n corresponds to one more half-wavelength fitting on the string.