Stationary waves

Stationary Waves Experiment for Schools, Teachers, and Students

Stationary waves (or standing waves) are waves that are formed when two waves of the same frequency and amplitude traveling in opposite directions superpose, resulting in a wave pattern that appears to be stationary. These waves exhibit nodes (points of no displacement) and antinodes (points of maximum displacement).

Theory:

1. Formation of Stationary Waves:

Stationary waves are formed due to the interference of two identical waves traveling in opposite directions.
For a wave traveling to the right:
y₁ = A sin(kx – ωt)
For a wave traveling to the left:
y₂ = A sin(kx + ωt)
The resultant wave is:
y = y₁ + y₂ = 2A sin(kx) cos(ωt)

where:

• A: Amplitude of each wave,
• k: Wave number (k = 2π / λ),
• ω: Angular frequency (ω = 2πf),
• t: Time, x: Position.

The resultant wave does not propagate; instead, it oscillates between fixed points.

2. Nodes and Antinodes:

• Nodes: Points where the displacement of the medium is always zero.
  Condition: sin(kx) = 0
  Node positions: x = n(λ / 2) (n = 0, 1, 2, …)
• Antinodes: Points where the displacement of the medium is maximum.
  Condition: sin(kx) = ±1
  Antinode positions: x = (n + ½)(λ / 2) (n = 0, 1, 2, …)

3. Characteristics of Stationary Waves:

• Stationary waves do not transfer energy from one point to another.
• Nodes remain stationary and have zero amplitude.
• Antinodes oscillate with maximum amplitude.
• The distance between two consecutive nodes or antinodes is λ / 2.
• The distance between a node and an adjacent antinode is λ / 4.

Types of Stationary Waves:

1. In Strings:

• Stationary waves in stretched strings are formed when waves are reflected at fixed ends.
• Boundary conditions: Both ends are nodes.
• Fundamental frequency:
  f₁ = v / 2L, where v = √(T / μ),
  v is the wave speed, T is the tension, and μ is the linear mass density.

2. In Pipes:

• Stationary waves in air columns occur in open or closed pipes.
• Open pipe: Both ends are antinodes.
• Closed pipe: One end is a node, and the other is an antinode.
• Fundamental frequency for open pipe:
  f₁ = v / 2L
• Fundamental frequency for closed pipe:
  f₁ = v / 4L

Applications of Stationary Waves:

• Used in musical instruments like stringed instruments (guitar, violin) and wind instruments (flute, trumpet).
• Understanding resonance phenomena in mechanical and acoustic systems.
• Used in the study of sound waves, vibrations, and harmonics.

Real-Life Examples:

• Vibrations in stretched strings of musical instruments.
• Formation of standing waves in organ pipes or flute tubes.
• Standing waves observed on the surface of water in a vibrating container.

Observations:

• The amplitude of the wave is zero at nodes and maximum at antinodes.
• The wave does not propagate; instead, energy is trapped between the boundaries.
• The distance between two consecutive nodes or antinodes is half the wavelength (λ / 2).
• Higher harmonics can form by increasing the frequency of the incident waves.