Resonance and standing waves

Forced oscillation

  • Produced when an external periodic driving force vibrates a system
  • The system will vibrate at the same frequency as the applied force
  • The system will continue to vibrate as force is applied

Free oscillation

  • When an object is struck, and then allowed to vibrate without further interference

Natural frequency

  • The frequency at which an object will naturally vibrate
  • In the absence of a driving force, objects will tend to vibrate at these frequencies


  • When forced frequency is similar to the natural frequency and applied in the same direction, resonance occurs
  • The amplitude of the vibration increases greatly as a result
  • The system hence oscillates with greater amplitude at some frequencies than others
  • The resonant frequency is the same as the natural frequency

Standing waves

  • Waves that appear to be standing still
  • Waves and reflected waves combine to form standing waves
  • The relative positions of the nodes and antinodes are the same, but the wave still travels.
  • Nodes are points of minimum displacement – half a wavelength apart
  • Antinodes are points of maximum displacement – also half a wavelength apart

Vibration in a string

  • In a string, the frequencies at which standing waves are established are called the natural frequency
  • Like all systems, it has more than one natural frequency, but prefers its fundamental
  • If f is the frequency of the first harmonic (n = 1), and L is the length of the string,

╬╗n = 2L/n

  • Frequency can be calculated using the wave equation, or fn = nf
  • The ends of the string are fixed so there is always zero displacement (i.e. nodes)

Open ended pipe

  • e.g. didgeridoo, flute
  • At open ends, sound waves are reflected with a phase change
  • This produces a pressure node at the ends of the pipe and a displacement antinodes

Closed ended pipe

  • e.g. clarinet
  • At closed ends, it reflects sound with no change of phase
  • At closed ends, there is a pressure antinode and a displacement node

Double wavelength of open end.


Open end
  • The air particles are free to move in or out
  • Hence max displacement but constant average pressure
  • Therefore displacement node and pressure antinode
Closed end
  • Air particles are pressed against the closed end and cannot move
  • Hence min displacement but max pressure
  • Therefore displacement node and pressure antinode

When a displacement node occurs, a pressure antinode does too, and vice versa.

  • ┬áLoud sound occurs at pressure antinodes (displacement nodes)
  • Soft sound occurs at pressure nodes (displacement antinodes)



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