Standing Waves and Resonance

• Standing Waves

  • Produced by two waves travelling in opposite directions
  • Superposition produces a wave pattern where the crests and troughs only move vertically
• Main differences between progressive and stationary waves

  • Standing waves store energy
  • Progressive waves transfer energy
• The principle of superposition applies to all types of waves i.e. transverse and longitudinal, progressive and stationary
• At the nodes

  Waves are in anti-phase causing destructive interference
• Standing waves store energy while progressive waves transfer energy
• Principle of superposition

  Two waves travelling in opposite directions along the same line with the same frequency superpose
• Formation of Nodes and Antinodes

  1. Nodes are locations of zero amplitude separated by half a wavelength (λ/2)

  2. Antinodes are locations of maximum amplitude

  3. Nodes are fixed and antinodes only oscillate in the vertical direction
• At the antinodes

  Waves are in phase causing constructive interference
• Simply blowing across the open end of a pipe can produce a standing wave in the pipe
• When a progressive wave travels towards a free end for a string, or open end for a pipe

  The reflected wave is in phase with the incident wave. The amplitudes of the incident and reflected waves add up. A free end is a location of maximum displacement - i.e. an antinode
• Phase difference on standing waves is different from travelling (progressive) waves
• Stationary waves can form on strings or in pipes
• Points with an even number of nodes between them


  Are in phase
• As progressive waves of different natural frequencies are sent along the string, standing waves with different numbers of nodes and antinodes form
• Phase on a Standing Wave

  Two points on a standing wave are either in phase or in anti-phase
• Points with an odd number of nodes between them


  Are in anti-phase
• Nodes and antinodes are a result of destructive and constructive interference respectively
• Constructive and destructive interference can be seen from the phase differences between two waves
• All points within a "loop"


  Are in phase
• For a pipe, there is more than one possible boundary condition: closed at both ends, open at both ends, closed at one end and open on the other
• At specific frequencies, known as natural frequencies, an integer number of half wavelengths will fit on the length of the string
• When the air within a pipe vibrates, longitudinal waves travel along the pipe
• Phase differences between two points on travelling waves can be anything from 0 to 2π. Between two points on a standing wave can only be in-phase (0 phase difference) or anti-phase (π out of phase)
• Harmonics on Strings


  • The simplest wave pattern is a single loop made up of two nodes and an antinode, called the first harmonic
  • The second harmonic has three nodes and two antinodes
  • The third harmonic has four nodes and three antinodes
• The general expression for the wavelength of the nth harmonic on a string that is fixed at both ends is: λn = 2L/n
• Standing wave inside a pipe open at both ends

  1. Wave reflection is in anti-phase with the incident wave

  2. The two waves cancel out

  3. An open end is a location of maximum displacement - i.e. an antinode
• For a pipe open at one end, only odd harmonics can exist under this boundary condition
• For a pipe open at both ends, the nth harmonic will have (n + 1) antinodes and n nodes
• Wave reflection towards a free end for a string, or open end for a pipe

  • The reflected wave is in phase with the incident wave
  • The amplitudes of the incident and reflected waves add up
  • A free end is a location of maximum displacement - i.e. an antinode
• The nth harmonic will have (n + 1) nodes and n antinodes
• Harmonics in Strings & Pipes

  • Stationary waves can have different wave patterns, known as harmonics
  • Harmonics depend on the frequency of the vibration and the boundary conditions (i.e. fixed and/or free ends)
  • Harmonics are the only frequencies and wavelengths that will form standing waves on strings or in pipes
• Forced Oscillations are produced by a periodic external force
• Calculating natural frequencies under both boundary conditions

  Natural frequencies are calculated from the wavelength of the standing wave and the speed v of the travelling waves using the wave equation
• When the driving frequency approaches the natural frequency of an oscillator, the system gains more energy from the driving force. Eventually, when they are equal, the oscillator vibrates at its maximum amplitude
• Resonance


  The frequency of the forced oscillations on a system is referred to as the driving frequency f. All oscillating systems have a natural frequency f which is defined as the frequency of an oscillation when the oscillating system is allowed to oscillate freely. Oscillating systems can exhibit a property known as resonance when the driving frequency f = natural frequency f
• Expression for the wavelength of the nth harmonic in a pipe of length L


  λn = 4L / n (where n is an odd number - i.e. 1, 3, 5...)
• Free Oscillations occur when there is no transfer of energy to or from the surroundings
• Every system has a fixed natural frequency
• Pushing a child on a swing is an example of how forced oscillations can produce resonance
• Damping


  The reduction in energy and amplitude of oscillations due to resistive forces on the oscillating system