• Interference?occurs whenever two or more waves combine to produce a resultant wave with a new resultant?displacement
• The waves combine according to the?principle of superposition
  • Constructive interference?happens when the resultant wave has a larger displacement than any of the individual displacements
  • Destructive interference?happens when the positive displacement of one wave and the negative displacement of another wave exactly cancel out giving a resultant displacement of zero

Coherence

• Interference is only observable if produced by a?coherent?source
• Waves are said to be coherent if they have:
  • A?constant?phase difference
  • The?same?frequency

Coherent waves (on the left) and non-coherent waves (on the right). The abrupt change in phase creates an inconsistent phase difference

• A coherent beam of light contains light waves that are?monochromatic?and have a constant phase difference
  • Monochromatic light consists of light waves of a?single frequency?

• Laser light is an example of a coherent light source
• Filament lamps produce incoherent light waves

Double-Slit Interference of Light

• When a?coherent beam of light?is incident on two narrow slits very close together,?diffraction?occurs at each slit (i.e. the waves spread out)
• As the diffracted waves cross, they?interfere?with each other
• If a screen is placed some distance away from the slits, a pattern of equally spaced bright and dark?fringes?is observed on the screen
  • The bright fringes form where the waves interfere constructively (i.e. a crest meets a crest or a trough meets a trough)
  • The dark fringes form where the waves interfere destructively (i.e. a crest meets a trough)

Coherent light waves interfere after passing through two narrow slits. Alternating bright and dark fringes are observed on the screen

• 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 is:
  • (6.5λ – 6λ) =?λ / 2?at point P₁
  • (7λ – 6λ) =?λ?at point P₂

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

Conditions for Constructive and Destructive Interference

• In general, for waves emitted by two coherent sources very close together:
  • The condition for?constructive interference?is:

path difference = nλ

• • The condition for?destructive interference?is:

path difference = (n + ?)λ

• • Where:
    • λ = wavelength of the waves in metres (m)
    • n = 0, 1, 2, 3... (any other integer)

• The same conditions apply to waves emitted by a single coherent source and diffracted by two narrow slits very close together

Path Difference and Wavefronts

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 (n?= 2), hence?constructive interference?occurs?at?point?P

Step 1: Count the number of wavelengths between each source and the desired point?

• • E.g. Number of wavelengths between:
    • S₁?? A = 5λ
    • S₂?? A?= 6.5λ

Step 2: Determine the path difference by subtracting the distances of the point from the two sources?

• • E.g. Path difference at A = (6.5λ – 5λ) = 1.5λ

Step 3: Compare the path difference calculated in Step 2 with the condition for constructive or destructive interference and give the value of?n

• • E.g. Path difference at A = 1.5λ =?(n + ?)λ ? n?= 1

Step 4: Decide whether the point is a location of constructive or destructive interference?

• • E.g. A is a location of destructive interference

• Point?A:
  • Path difference = (6.5λ – 5λ) = 1.5λ
  • n?= 1
  • Destructive interference
• Point?B:
  • Path difference = (5λ – 4λ) = λ
  • n?= 1
  • Constructive interference
• Point?C:
  • Path difference = (2λ – 2λ) = 0
  • n?= 0
  • Constructive interference
• Point?D:
  • Path difference = (5λ – 4.5λ) = 0.5λ
  • n?= 0
  • Destructive interference
• Point?E:
  • Path difference = (4λ – 3λ) = λ
  • n?= 1
  • Constructive interference