Standing Waves & Resonance

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Notes de leçon

Standing Waves

  • Standing waves (or stationary waves) are produced when two waves travelling in opposite directions along the same line with the same frequency superpose.
  • This usually occurs when a travelling wave superimposes with its reflection.
  • For a standing wave to form, the two waves must have the same wavelength and similar amplitude.
  • In a standing wave, the crests and troughs only move vertically; the wave pattern does not travel along the medium.
  • Standing waves store energy, whereas progressive (travelling) waves transfer energy.
  • The principle of superposition applies to all types of waves, including transverse and longitudinal waves.

Nodes & Antinodes

  • Nodes are locations of zero amplitude, separated by half a wavelength (λ/2).
  • Antinodes are locations of maximum amplitude.
  • Nodes and antinodes do not move along the wave; nodes are fixed and antinodes oscillate vertically.
  • At nodes, the two waves are in anti-phase (crest meets trough), causing destructive interference and cancellation.
  • At antinodes, the two waves are in phase (crest meets crest), causing constructive interference and addition.
  • Two points on a standing wave are either in phase (0 phase difference) or in anti-phase (π out of phase).
  • Points with an odd number of nodes between them are in anti-phase; points with an even number of nodes between them are in phase.
  • All points within a single loop are in phase.

Boundary Conditions for Standing Waves

  • Standing waves form on strings or in pipes when progressive waves superimpose with their reflections.
  • The number of nodes and antinodes that fit depends on the frequency of the progressive waves and the boundary conditions (fixed/free ends).
  • For a string, boundary conditions can be: fixed at both ends, free at both ends, or one end fixed and the other free.
  • At a free end (or open pipe end), the reflected wave is in phase with the incident wave, producing an antinode (maximum displacement).
  • At a fixed end (or closed pipe end), the reflected wave is in anti-phase with the incident wave, producing a node (zero displacement).
  • For a pipe, possible boundary conditions are: closed at both ends, open at both ends, or closed at one end and open at the other.
  • When air vibrates in a pipe, longitudinal waves travel along the pipe and can form standing waves.

Harmonics in Strings & Pipes

  • Harmonics are the only frequencies and wavelengths that form standing waves on strings or in pipes, depending on boundary conditions.
  • For a string fixed at both ends: the nth harmonic has (n + 1) nodes and n antinodes.
  • The wavelength of the nth harmonic on a string fixed at both ends is λₙ = 2L/n, where L is the string length.
  • The frequency of the nth harmonic on a string is fₙ = nv/(2L), where v is the wave speed.
  • For a pipe open at both ends: the nth harmonic has n nodes and (n + 1) antinodes; the same expressions λₙ = 2L/n and fₙ = nv/(2L) apply.
  • For a pipe open at one end: only odd harmonics exist, with wavelength λₙ = 4L/n, where n is an odd integer (1, 3, 5...).
  • The first harmonic (fundamental) on a string fixed at both ends has one loop, two nodes, and one antinode, with λ₁ = 2L.
  • The second harmonic on a string has three nodes and two antinodes; the third harmonic has four nodes and three antinodes.

The Nature of Resonance

  • Free oscillations occur when there are only internal forces acting and no energy input; the system oscillates at its natural frequency.
  • Forced oscillations are produced by a periodic external driving force, causing the system to oscillate at the driving frequency.
  • The natural frequency f₀ is the frequency at which a system oscillates freely.
  • Resonance occurs when the driving frequency equals the natural frequency (f = f₀), resulting in maximum amplitude.
  • At resonance, energy is transferred most efficiently from the driver to the oscillating system.
  • If the driving frequency is slightly lower or higher than the natural frequency, the amplitude increases but to a lesser extent.
  • A child on a swing is an example of resonance: pushing at the natural frequency increases the amplitude.

The Effect of Damping

  • Damping is the reduction in energy and amplitude of oscillations due to resistive forces (e.g., friction, air resistance).
  • Damping does not change the frequency (or period) of the oscillations; the amplitude decreases over time.
  • Light damping: amplitude decays exponentially with time, and the system oscillates with gradually decreasing amplitude.
  • Critical damping: the system returns to equilibrium in the shortest possible time without oscillating.
  • Heavy damping: the system returns to equilibrium slowly without oscillating, taking longer than critical damping.
  • On a resonance curve (amplitude vs driving frequency), increased damping lowers the peak, broadens the curve, and may shift the peak slightly left.
  • Damping reduces the sharpness of resonance and the amplitude at the resonant frequency, but the natural frequency remains the same.

Diapos

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  1. 1.What is the phase relationship between the two waves at a node of a standing wave?

    Easy
    • AThey are in anti-phase, so destructive interference occurs
    • BThey are in phase, so constructive interference occurs
    • CThey are in quadrature (90° out of phase)
    • DThey have a phase difference that varies with time

  2. 2.A standing wave is set up on a string fixed at both ends. The distance between two consecutive nodes is 0.25 m. What is the wavelength of the standing wave?

    Medium
    • A0.125 m
    • B0.25 m
    • C0.50 m
    • D1.00 m

  3. 3.A pipe is open at one end and closed at the other. Which of the following frequencies could NOT form a standing wave in this pipe?

    Medium
    • A150 Hz
    • B300 Hz
    • C450 Hz
    • D750 Hz

  4. 4.A standing wave is formed in a pipe of length L that is open at both ends. Which expression gives the wavelength of the second harmonic?

    Medium
    • Aλ = 2L
    • Bλ = L
    • Cλ = L/2
    • Dλ = 4L

  5. 5.A standing wave is formed on a stretched string. Which of the following correctly describes the energy of the wave?

    Medium
    • AIt transfers energy along the string
    • BIt stores energy and does not transfer it along the string
    • CIt transfers energy but at half the speed of a travelling wave
    • DIt loses all its energy at the nodes

  6. 6.The frequency of a damped oscillation changes as the amplitude decreases.

    Easy

    True or false?

  7. 7.In a pipe that is open at one end, only odd harmonics can be formed.

    Easy

    True or false?

  8. 8.Match each term with its correct definition.

    Medium
    • Node
    • Antinode
    • Resonance
    • Damping
    • A point of zero amplitude on a standing wave
    • A point of maximum amplitude on a standing wave
    • When the driving frequency equals the natural frequency, giving maximum amplitude
    • The reduction in energy and amplitude of oscillations due to resistive forces

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