Where waves meet.
When waves overlap, their displacements add. Equal in-phase waves reinforce; equal opposite displacements cancel momentarily. Reflections can create standing waves with fixed nodes and antinodes. Adjacent nodes are half a wavelength apart.
For a string fixed at both ends, allowed wavelengths satisfy L = nλ/2. Resonance produces a large response when a driving frequency approaches a natural frequency; damping limits the amplitude. Real measurements need uncertainty estimates and repeated trials before a model can be evaluated.
A string fixed at both ends
Explore the relationship
A 1 m string. The graph is one maximum-displacement snapshot; nodes stay fixed as the string oscillates.
Tap or focus a plotted point to read its values.
View plotted data as a table
| Series | Position / m | Displacement envelope / m |
|---|---|---|
| Displacement envelope / m | 0 | 0 |
| Displacement envelope / m | 0.0166667 | 0.104528 |
| Displacement envelope / m | 0.0333333 | 0.207912 |
| Displacement envelope / m | 0.05 | 0.309017 |
| Displacement envelope / m | 0.0666667 | 0.406737 |
| Displacement envelope / m | 0.0833333 | 0.5 |
| Displacement envelope / m | 0.1 | 0.587785 |
| Displacement envelope / m | 0.116667 | 0.669131 |
| Displacement envelope / m | 0.133333 | 0.743145 |
| Displacement envelope / m | 0.15 | 0.809017 |
| Displacement envelope / m | 0.166667 | 0.866025 |
| Displacement envelope / m | 0.183333 | 0.913545 |
| Displacement envelope / m | 0.2 | 0.951057 |
| Displacement envelope / m | 0.216667 | 0.978148 |
| Displacement envelope / m | 0.233333 | 0.994522 |
| Displacement envelope / m | 0.25 | 1 |
| Displacement envelope / m | 0.266667 | 0.994522 |
| Displacement envelope / m | 0.283333 | 0.978148 |
| Displacement envelope / m | 0.3 | 0.951057 |
| Displacement envelope / m | 0.316667 | 0.913545 |
| Displacement envelope / m | 0.333333 | 0.866025 |
| Displacement envelope / m | 0.35 | 0.809017 |
| Displacement envelope / m | 0.366667 | 0.743145 |
| Displacement envelope / m | 0.383333 | 0.669131 |
| Displacement envelope / m | 0.4 | 0.587785 |
| Displacement envelope / m | 0.416667 | 0.5 |
| Displacement envelope / m | 0.433333 | 0.406737 |
| Displacement envelope / m | 0.45 | 0.309017 |
| Displacement envelope / m | 0.466667 | 0.207912 |
| Displacement envelope / m | 0.483333 | 0.104528 |
| Displacement envelope / m | 0.5 | 1.22465e-16 |
| Displacement envelope / m | 0.516667 | -0.104528 |
| Displacement envelope / m | 0.533333 | -0.207912 |
| Displacement envelope / m | 0.55 | -0.309017 |
| Displacement envelope / m | 0.566667 | -0.406737 |
| Displacement envelope / m | 0.583333 | -0.5 |
| Displacement envelope / m | 0.6 | -0.587785 |
| Displacement envelope / m | 0.616667 | -0.669131 |
| Displacement envelope / m | 0.633333 | -0.743145 |
| Displacement envelope / m | 0.65 | -0.809017 |
| Displacement envelope / m | 0.666667 | -0.866025 |
| Displacement envelope / m | 0.683333 | -0.913545 |
| Displacement envelope / m | 0.7 | -0.951057 |
| Displacement envelope / m | 0.716667 | -0.978148 |
| Displacement envelope / m | 0.733333 | -0.994522 |
| Displacement envelope / m | 0.75 | -1 |
| Displacement envelope / m | 0.766667 | -0.994522 |
| Displacement envelope / m | 0.783333 | -0.978148 |
| Displacement envelope / m | 0.8 | -0.951057 |
| Displacement envelope / m | 0.816667 | -0.913545 |
| Displacement envelope / m | 0.833333 | -0.866025 |
| Displacement envelope / m | 0.85 | -0.809017 |
| Displacement envelope / m | 0.866667 | -0.743145 |
| Displacement envelope / m | 0.883333 | -0.669131 |
| Displacement envelope / m | 0.9 | -0.587785 |
| Displacement envelope / m | 0.916667 | -0.5 |
| Displacement envelope / m | 0.933333 | -0.406737 |
| Displacement envelope / m | 0.95 | -0.309017 |
| Displacement envelope / m | 0.966667 | -0.207912 |
| Displacement envelope / m | 0.983333 | -0.104528 |
| Displacement envelope / m | 1 | -2.44929e-16 |
Explore: Change the harmonic number and count nodes and antinodes.
A 0.60 m string has wave speed 120 .
- Fundamental wavelength = 2L = 1.20 m.
- Fundamental frequency = = 100 Hz.
Assumed knowledge
Wave speed and frequency.
Learning checkpoints & source
YOUR LEARNING CHECKPOINT- Apply superposition and identify standing waves.
- Explain resonance and experimental uncertainty.