3.3.1 Path Difference & Coherence

• Interference occurs when waves overlap and their resultant displacement is the sum of the displacement of each wave
• This result is based on the principle of superposition and the resultant waves may be smaller or larger than either of the two individual waves

Coherence

• At points where the two waves are neither in phase nor in antiphase, the resultant amplitude is somewhere in between the two extremes
• Waves are said to be coherent if they have:
  • The same frequency
  • A constant phase difference

Coherent v non-coherent wave. The abrupt change in phase creates an inconsistent phase difference

• Coherence is vital in order to produce an observable, or hearable, interference pattern
  • Laser light is an example of a coherent light source, whereas filament lamps produce incoherent light waves
  • When coherent sound waves are in phase, the sound is louder because of constructive interference

Path Difference

• Path difference is defined as:

The difference in distance travelled by two waves from their sources to the point where they meet

• Path difference is generally expressed in multiples of wavelength

At point P₂ the waves have a path difference of a whole number of wavelengths resulting in constructive interference. At point P₁ the waves have a path difference of an odd number of half wavelengths resulting in destructive interference 

• In the diagram above, the number of wavelengths between:
  • S₁ ➜ P₁ = 6λ
  • S₂ ➜ P₁ = 6.5λ
  • S₁ ➜ P₂ = 7λ
  • S₂ ➜ P₂ = 6λ

• The path difference at point P₁ is 6.5λ – 6λ = λ / 2
• The path difference at point P₂ is 7λ – 6λ = λ

• In general:
  • The condition for constructive interference is a path difference of nλ
  • The condition for destructive interference is a path difference of (n + ½)λ
  • In this case, n is an integer i.e. 0, 1, 2, 3...

At point P the waves have a path difference of a whole number of wavelengths resulting in constructive interference

• Another way to represent waves spreading out from two sources is shown in the diagram above
• At point P, the number of crests from:
  • Source S₁ = 4λ
  • Source S₂ = 6λ

• The path difference at P is 6λ – 4λ = 2λ
• This is a whole number of wavelengths, hence constructive interference occurs at point P

An experiment demonstrating interference of water waves.

Water ripples are generated from two red point sources. As the waves travel away from their source they interfere with each other. When the peaks from each wave meet at the same blue point constructive interference occurs and the water is highest. When the troughs from each wave meet at the same orange point constructive interference occurs and the water is most shallow. When the peak from one wave meets the trough of another destructive interference occurs and the water here is flat. 

      ANSWER: B

• At point X:
  • Both peaks of the waves are overlapping
  • Path difference = 5.5λ – 4.5λ = λ
  • This is constructive interference and rules out options C and D

• At point Y:
  • Both troughs are overlapping
  • Path difference = 3.5λ – 3.5λ = 0
  • Therefore constructive interference occurs

• • A peak of one of the waves meets the trough of the other
  • Path difference = 4λ – 3.5λ = λ / 2
  • This is destructive interference
  • At point Z: