Stationary Waves: Experiments and Concepts
1. What are Stationary Waves? 🎶
A stationary wave is a pattern that does not travel through a medium. Instead, points on the wave called nodes stay still, while other points called antinodes move back and forth. Think of a tug‑of‑war rope that stays in place while the rope itself moves up and down. The key equations are:
- \$v = f \lambda\$ – wave speed equals frequency times wavelength.
- \$f = \frac{n v}{2L}\$ – for a string fixed at both ends, only certain frequencies (harmonics) fit, where \$n\$ is an integer (1, 2, 3, …).
- \$f = \frac{n v}{4L}\$ – for a pipe open at one end and closed at the other, only odd harmonics (1, 3, 5, …) fit.
2. Stretched String Experiment 🎸
A string stretched between two fixed points behaves like a guitar string. By plucking or driving it at different frequencies, we can see which frequencies produce clear standing waves.
- Set the string length \$L\$ (e.g., 1.0 m).
- Apply a known tension \$T\$ (e.g., 200 N).
- Use a tuning fork or an electronic oscillator to drive the string at a frequency \$f\$.
- Observe the string with a high‑speed camera or a strobe light to identify nodes and antinodes.
- Record the fundamental frequency \$f1\$ and higher harmonics \$f2, f_3, …\$.
The wave speed on the string is \$v = \sqrt{T/\mu}\$, where \$\mu\$ is the mass per unit length. Using \$f = \frac{n v}{2L}\$, you can predict the expected frequencies and compare them with your measurements.
| Harmonic (n) | Predicted \$f_n\$ (Hz) | Measured \$f_n\$ (Hz) |
|---|
| 1 | \$f_1 = \frac{v}{2L}\$ | — |
| 2 | \$f_2 = \frac{2v}{2L}\$ | — |
| 3 | \$f_3 = \frac{3v}{2L}\$ | — |
3. Air Column Experiment 🎤
An air column inside a tube behaves like a musical instrument. By changing the length of the column, we can observe different standing wave patterns.
- Use a tube of length \$L\$ (e.g., 0.5 m) that is open at both ends.
- Place a tuning fork or a speaker at one end to produce a sound of frequency \$f\$.
- Move a hand or a movable end stop along the tube to change the effective length \$L_{\text{eff}}\$.
- Listen for the loudest (resonant) sounds – these correspond to antinodes at the open ends.
- Record the resonant lengths \$L_{\text{eff}}\$ for the first few harmonics.
For a tube open at both ends, the condition for resonance is \$L{\text{eff}} = n \frac{\lambda}{2}\$, leading to \$fn = \frac{n v}{2L_{\text{eff}}}\$ with \$n = 1, 2, 3, …\$. If the tube were closed at one end, only odd \$n\$ would appear.
4. Microwave Standing Wave Experiment 📡
Microwaves can create standing waves inside a rectangular cavity or a waveguide. This experiment is a bit more advanced but illustrates the same physics.
- Place a microwave source (like a small microwave oven or a signal generator) near a metallic cavity.
- Use a probe or a small antenna to detect the electric field intensity inside the cavity.
- Move the probe along the cavity’s length and record the points where the field is minimal (nodes) and maximal (antinodes).
- Measure the distance between successive nodes – this is half the wavelength \$\lambda/2\$.
- Calculate the wavelength \$\lambda\$ and, using the known frequency \$f\$, verify \$v = f \lambda\$ (for microwaves \$v\$ is close to the speed of light \$c\$).
Because microwaves travel in a vacuum or air, the end corrections are negligible, simplifying the analysis. The pattern you see is exactly the same as the guitar string or flute, just at a much higher frequency!
5. Key Takeaways 📚
- Stationary waves arise when waves of the same frequency and amplitude travel in opposite directions and interfere.
- Nodes are fixed points; antinodes oscillate with maximum amplitude.
- For a string fixed at both ends, only harmonics \$n = 1, 2, 3, …\$ satisfy \$L = n \frac{\lambda}{2}\$.
- For an open air column, the same condition holds; for a closed column, only odd harmonics appear.
- Microwave standing waves confirm the universality of the wave equation across different media.
- End corrections are small in these experiments, so the simple formulas above give accurate predictions.