Standing Wave
A standing wave is a wave pattern formed by the superposition of two identical waves travelling in opposite directions, resulting in a stationary pattern of nodes (zero displacement) and antinodes (maximum displacement). Standing waves do not transport energy along the medium; instead, energy oscillates between kinetic and potential forms at fixed positions. They are fundamental to the physics of musical instruments, laser cavities, and microwave resonators.
Key Formula
f_n = (n × v) / (2 × L)
LaTeX: f_n = \dfrac{n v}{2L}
| Symbol | Meaning | Unit |
|---|---|---|
| f_n | Frequency of the nth harmonic | Hz |
| n | Harmonic number (1, 2, 3, …) | dimensionless |
| v | Wave speed in the medium | m/s |
| L | Length of the string or pipe | m |
Worked Example
Problem
A guitar string of length 0.65 m has a wave speed of 390 m/s. Find the fundamental frequency (n = 1) and the second harmonic (n = 2).
Solution
Step 1: Use f_n = n·v / (2L). Step 2: Fundamental (n=1): f₁ = 1 × 390 / (2 × 0.65) = 390 / 1.3 = 300 Hz. Step 3: Second harmonic (n=2): f₂ = 2 × 390 / 1.3 = 600 Hz.
Answer
f₁ = 300 Hz; f₂ = 600 Hz
Harmonics of a Standing Wave on a String
| Harmonic (n) | Number of Nodes | Number of Antinodes | Frequency |
|---|---|---|---|
| 1 (fundamental) | 2 | 1 | f₁ |
| 2 (2nd harmonic) | 3 | 2 | 2f₁ |
| 3 (3rd harmonic) | 4 | 3 | 3f₁ |
| 4 (4th harmonic) | 5 | 4 | 4f₁ |
| n | n + 1 | n | nf₁ |
Interactive Tools
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Related Terms
Node (wave)
A node is a point in a standing wave where the displacement of the medium is permanently zero due to destructive interference between the two superposed waves. Nodes remain stationary regardless of time, and they occur at intervals of half a wavelength (λ/2) along the medium. In a vibrating string fixed at both ends, the fixed endpoints are always nodes, and the number of nodes determines the harmonic mode of vibration.
Antinode
An antinode is a point in a standing wave where the displacement of the medium reaches its maximum amplitude, caused by constructive interference between the two superposed waves. Antinodes oscillate with the greatest energy in the standing wave pattern and are located exactly halfway between consecutive nodes, at intervals of half a wavelength. They are the positions of maximum vibration in musical instrument strings and air columns.
Wave
A wave is a disturbance that transfers energy through a medium or through space without permanently displacing the medium itself. Waves are fundamental to how energy propagates in nature, from ocean ripples to light traveling across the universe. They are characterised by properties such as wavelength, frequency, amplitude, and speed, and underpin technologies ranging from radio communication to medical ultrasound.
The term "standing" reflects the wave's stationary appearance — unlike a travelling wave, the pattern does not move forward. First analysed mathematically by Leonhard Euler and others in the 18th century.