# 3.3.1 Path Difference & Coherence

### 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
• The condition for destructive interference is a path difference of (n + ½)λ
• In this case, n is an integer i.e. 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 =
• This is a whole number of wavelengths, hence constructive interference occurs at point P

#### Worked Example

The diagram shows the interferences of coherent waves from two point sources.

Which row in the table correctly identifies the type of interference at points X, Y and Z.

• 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

#### Exam Tip

Remember, interference of two waves can either be:

• In phase, causing constructive interference. The peaks and troughs line up on both waves. 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

Think of ‘constructive’ interference as ‘building’ the wave and ‘destructive’ interference as ‘destroying’ the wave.

### Author: Katie

Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.
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