5.8 Phase & Path Difference

• Waves are said to be coherent if they have:
  • The same frequency
  • A constant phase difference

Phase Difference

• Two points on a wave, or on different waves, are in phase when they are the same point in their wave cycle
• The angle between their wave cycles is the phase difference

Path Difference

• The type of interference occurring at a given point (i.e. constructive or destructive) depends on the path difference of the overlapping waves
• 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...

• Hence:
  • Destructive interference occurs at point P₁
  • Constructive interference occurs at point P₂

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

      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

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