4.9.3 Interference

Interference

• 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
  • The resultant waves may be smaller or larger than either of the two individual waves

• 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

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 a wavelength

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

Phase Difference

• Two waves with a path difference will also have a difference in phase
  • This is their phase difference

• Phase difference is defined as:

The difference in phase between two waves that arrive at the same point

• It is given as an angle, in radians or degrees

• Whether two waves will constructively or destructively interfere at a point is determined by its path difference or phase difference

Path Difference

• Path difference is determined in multiples of a wavelength
• Constructive interference occurs when there is a path difference of nλ
  • For example, 2λ

• Destructive interference occurs when there is a path difference of (n + ½)λ
  • For example, 3λ / 2 or 1.5λ

• In this case, n is an integer i.e. 1, 2, 3...

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
  • Therefore, this is destructive interference (half-wavelength difference)

• The path difference at point P₂ is 7λ – 6λ = λ
  • Therefore, this is constructive interference (a whole number of wavelengths difference)

Phase Difference

• The phase difference between two waves is determined by an angle, in radians or degrees
• Constructive interference occurs when the phase difference is an even multiple of π or that they are in phase
  • Eg. 2π, 4π

• Destructive interference occurs when the phase difference is an odd multiple of π or that they are in anti-phase
  • Eg. π, 3π

      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