The Principle of Superposition

• The principle of superposition states:

When two or more waves with the same frequency arrive at a point, the resultant displacement is the sum of the displacements of each wave

• This principle describes how waves that meet at a point in space interact
• When two waves with the same frequency and amplitude arrive at a point, they superpose either:
  • In phase, causing?constructive interference.?The peaks and troughs line up on both waves and the resultant wave has double the amplitude
  • In?anti-phase, causing?destructive interference.?The peaks on one wave line up with the troughs of the other. The resultant wave has no amplitude

   

Waves in superposition can undergo constructive or destructive interference

• The principle of superposition applies to all types of waves i.e. transverse and longitudinal, progressive and stationary

The Formation of Stationary Waves

• A stationary wave is formed when:

Two waves travelling in opposite directions along the same line with the same frequency superpose

• The waves must have:
  • The same?wavelength)
  • A similar amplitude

   

• As a result of superposition, a resultant wave is produced

Nodes and antinodes are a result of destructive and constructive interference respectively

• At the?nodes:
  • The waves are in anti-phase meaning destructive interference occurs
  • This causes the two waves to cancel each other out

   

• At the?antinodes:
  • The waves are in phase meaning constructive interference occurs
  • This causes the waves to add together

   

• Each point on the stationary wave has a?different?amplitude (unlike a progressive / travelling wave where each point has the?same?amplitude)

A graphical representation of how stationary waves are formed - the black line represents the resulting wave

Stretched Strings

• Vibrations caused by stationary waves on a stretched string produce?sound
  • This is how stringed instruments, such as guitars or violins, work

   

• This can be demonstrated by a length of string under tension fixed at one end and vibrations made by an oscillator:

Stationary wave on a stretched string

• At specific frequencies, known as?resonant frequencies, a whole number of half wavelengths will fit on the length of the string
• As the resonant frequencies of the oscillator are achieved, standing waves with different numbers of minima (nodes) and maxima (antinodes) form

Microwaves

• A microwave source is placed in line with a reflecting plate and a small detector between the two
• The reflector can be moved to and from the source to vary the stationary wave pattern formed
• By moving the detector, it can pick up the minima (nodes) and maxima (antinodes) of the stationary wave pattern

Using microwaves to demonstrate stationary waves

Sound Waves

• Sound waves can be produced as a result of the formation of stationary waves inside an air column
  • This is how musical instruments, such as clarinets and organs, work

   

• This can be demonstrated by placing a fine powder inside the air column and a loudspeaker at the open end
• At certain frequencies, the powder forms evenly spaced heaps along the tube, showing where there is zero disturbance as a result of the nodes of the stationary wave

Stationary wave in an air column

• In order to produce a stationary wave, there must be a minima (node) at one end and a maxima (antinode) at the end with the loudspeaker