Waves & Sound
Standing wave on a string
Press Play — nodes (grey) stay still; antinodes (bright) oscillate with y = A sin(nπx/L) cos(ωt).
Create standing waves on a string and explore nodes, antinodes, and resonance. Adjust frequency and tension to match harmonics with live wavelength and wave speed data.
Standing wave
Reflections on a fixed string create nodes and antinodes — the pattern oscillates in place.
A standing wave is a superposition of two identical travelling waves going opposite directions. Nodes have zero amplitude; antinodes oscillate maximally.
- y = A sin(nπx/L) cos(ωt)
- λₙ = 2L/n
Investigation brief
Plan the question before you open the lab
The brief mirrors the prerendered page: driving question, competing predictions, variable roles, governing laws, setup, analysis and extension prompts remain visible and in this order.
Driving question
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?
Predictions to weigh
- Harmonic frequencies follow no simple pattern relative to the fundamental.
- Each harmonic's frequency is half the previous one.
- Each harmonic's frequency is a whole-number multiple of the fundamental.
Variable roles
What you set:
- Harmonic number n
What you measure:
- Fundamental f₁ (Hz)
- Resonant frequency fₙ (Hz)
How the investigation runs
- 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.
Governing equation
Fixed-End Standing-Wave Fundamental — f₁ = v/(2L)
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 — fₙ = n·f₁
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.
What the printable worksheet asks students to work out
- 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.
Where this shows up beyond the lab
- 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.
- AP Physics 2 — Unit 14: Waves, Sound, and Physical Optics
- IB Physics — C.4 Standing waves and resonance
- General High School Physics — Waves & sound
- NGSS High School Physics — Wave properties
- What pattern?
- Raise the harmonic
- Capture the resonance
- Open the data
- Explain your evidence
Open the interactive simulation to build the scene, press Play, and explore with live measurements and a guided tutorial.