Standing Waves and Resonance
Standing Waves and Resonance
A standing wave (stationary wave) is formed when two progressive waves of the same frequency, wavelength, and amplitude travel in opposite directions and superpose. Unlike progressive waves, standing waves do not transfer energy along the medium.
Formation of Standing Waves
Standing waves are most commonly formed by:
- A wave reflecting back on itself (e.g., a string fixed at both ends)
- Two speakers facing each other emitting the same frequency
At certain points, the waves always arrive in phase and constructive interference produces maximum amplitude — these are antinodes.
At other points, the waves always arrive in antiphase (180° out of phase) and destructive interference produces zero amplitude — these are nodes.
Key Properties of Standing Waves
| Property | Standing Wave | Progressive Wave |
|---|---|---|
| Energy transfer | None | Energy transferred in direction of travel |
| Amplitude | Varies from zero (node) to maximum (antinode) | Same for all points |
| Phase | All points between adjacent nodes are in phase | Phase changes continuously along the wave |
| Wavelength | Distance between alternate nodes = λ | Distance between adjacent points in phase = λ |
| Frequency | All points vibrate at the same frequency | All points vibrate at the same frequency |
Important: The distance between adjacent nodes (or adjacent antinodes) is λ/2.
Standing Waves on Strings
For a string of length L fixed at both ends, standing waves can only form at specific frequencies where nodes exist at both fixed ends.
Fundamental mode (1st harmonic): L = λ₁/2 → λ₁ = 2L
The fundamental frequency f₁ = v/λ₁ = v/(2L)
where v is the wave speed on the string: v = √(T/μ)
- T = tension in the string (N)
- μ = mass per unit length (kg m⁻¹)
So the fundamental frequency is: f₁ = (1/2L)√(T/μ)
Harmonics:
- 2nd harmonic: f₂ = 2f₁, L = λ₂ (1 node in middle)
- 3rd harmonic: f₃ = 3f₁, L = 3λ₃/2 (2 nodes)
- nth harmonic: fₙ = nf₁, with (n−1) internal nodes
Standing Waves in Air Columns
Closed pipe (closed at one end, open at the other):
- Node at closed end, antinode at open end
- Fundamental: L = λ/4 → f₁ = v/(4L)
- Only odd harmonics exist: f₁, 3f₁, 5f₁, ...
Open pipe (open at both ends):
- Antinode at both ends
- Fundamental: L = λ/2 → f₁ = v/(2L)
- All harmonics exist: f₁, 2f₁, 3f₁, ...
Worked Example
A guitar string is 0.64 m long with mass per unit length 3.5 × 10⁻³ kg m⁻¹ and tension 73 N. Find the fundamental frequency.
v = √(T/μ) = √(73 / 3.5 × 10⁻³) = √(20857) = 144.4 m s⁻¹
f₁ = v/(2L) = 144.4 / (2 × 0.64) = 112.8 Hz
Resonance
Resonance occurs when the driving frequency of an external force matches the natural frequency of a system. At resonance, energy transfer from the driver to the system is most efficient, and the amplitude of oscillation reaches a maximum.
Conditions for Resonance
1. The system must have a natural frequency of vibration
2. An external periodic driving force must act on the system
3. The driving frequency must equal the natural frequency
4. The phase difference between driver and driven system is 90° at resonance
Amplitude-Frequency Response
The resonance curve shows how amplitude varies with driving frequency:
- At low frequencies: amplitude is small, oscillation approximately in phase with driver
- At resonance (f = f₀): amplitude is maximum, phase difference = π/2 (90°)
- At high frequencies: amplitude decreases, oscillation approximately in antiphase with driver
Damping and Resonance
Damping is the loss of energy from an oscillating system, usually to the surroundings as thermal energy.
- Light damping: amplitude decreases gradually over many oscillations; sharp resonance peak
- Heavy damping: amplitude decreases rapidly; broad, flattened resonance peak at lower amplitude
- Critical damping: system returns to equilibrium in the shortest time without oscillating
- Overdamping: system returns to equilibrium without oscillating but more slowly than critical damping
Increasing damping:
- Reduces the maximum amplitude at resonance
- Broadens the resonance curve
- Shifts the resonance frequency slightly below f₀
Examples of Resonance
- Barton's pendulums: a driver pendulum causes pendulums of matching length to oscillate with greatest amplitude
- Tacoma Narrows Bridge (1940): wind-driven oscillations matched the bridge's natural frequency
- Microwave ovens: the microwave frequency (2.45 GHz) matches the rotational frequency of water molecules
- MRI scanners: radio frequency pulses at the Larmor frequency cause hydrogen nuclei to resonate
Experimental Investigation: Melde's Experiment
A string attached to a vibration generator can demonstrate standing waves:
1. Adjust the frequency until standing wave patterns appear
2. Measure the wavelength from node spacing (λ = 2 × node separation)
3. Verify f × λ = v
4. Changing tension changes v and therefore the frequencies at which standing waves form
End Correction
In air column experiments, the antinode at an open end forms slightly beyond the physical end of the pipe. The end correction c is approximately 0.6 times the radius of the pipe:
Effective length = L + c (closed pipe) or L + 2c (open pipe)