• 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 + ?)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