Standing Waves
IntermediateExplore standing wave patterns, nodes, antinodes, and harmonics in vibrating strings and air columns.
💡 How to Use:
- Harmonic number: Higher harmonics have more nodes and shorter wavelengths
- String: Both ends fixed - supports all harmonics (n = 1, 2, 3, ...)
- Open pipe: Both ends open - supports all harmonics (antinodes at ends)
- Closed pipe: One end closed - only odd harmonics (n = 1, 3, 5, ...)
- Nodes: Points that never move (destructive interference)
- Antinodes: Points of maximum motion (constructive interference)
- Watch how the wave pattern changes while the nodes remain stationary!
Theory
Standing Waves Formation
Standing waves form when two waves of the same frequency and amplitude traveling in opposite directions interfere with each other, creating stationary patterns of nodes and antinodes.
String (Both Ends Fixed): L = n(λ/2), n = 1, 2, 3, ...
Frequency: fn = nv / (2L)
Wavelength: λn = 2L / n
Wave Speed: v = √(T/μ) for strings
Nodes and Antinodes
Nodes
Points of zero displacement where destructive interference always occurs. The medium remains stationary at these points.
- • Always at fixed boundaries
- • Number of nodes = n + 1 for nth harmonic
- • Separated by λ/2
Antinodes
Points of maximum displacement where constructive interference occurs. The medium oscillates with maximum amplitude.
- • Located between nodes
- • Number of antinodes = n for nth harmonic
- • Separated by λ/2
Harmonics
Harmonics are the natural frequencies at which a system can vibrate. Each harmonic corresponds to a specific standing wave pattern.
| Harmonic | Nodes | Antinodes | Wavelength | Frequency |
|---|---|---|---|---|
| 1st (Fundamental) | 2 | 1 | 2L | f₁ |
| 2nd | 3 | 2 | L | 2f₁ |
| 3rd | 4 | 3 | 2L/3 | 3f₁ |
| nth | n+1 | n | 2L/n | nf₁ |
Boundary Conditions
- String (both ends fixed): Nodes at both ends, supports all harmonics
- Pipe (both ends open): Antinodes at both ends, supports all harmonics
- Pipe (one end closed): Node at closed end, antinode at open end, only odd harmonics
- Free end: Antinode forms at free boundaries
Resonance
Resonance occurs when a system is driven at one of its natural frequencies (harmonics). This causes a dramatic increase in amplitude as energy efficiently transfers to the system.
Real-World Applications
- Musical instruments: Guitar strings, violin, piano - all use standing waves to produce sound
- Wind instruments: Flutes, organs, trumpets use standing waves in air columns
- Microwave ovens: Standing electromagnetic waves heat food
- Laser cavities: Standing light waves between mirrors
- Radio antennas: Designed to resonate at specific frequencies
- Earthquake engineering: Buildings can experience standing wave patterns during seismic events
- Acoustic treatment: Understanding nodes helps in soundproofing and studio design