Path Difference & Coherence (AQA A Level Physics): Revision Note

Path difference & coherence

Superposition and interference

• When two or more waves arrive at the same point and overlap, their amplitudes combine  

  • This is called superposition

• The principle of superposition states that:

When two or more waves overlap at a point, the displacement at that point is equal to the sum of the displacements of the individual waves

• The superposition of surface water waves shows the effect of this overlap

  • There are areas of zero displacement, where the water is flat

  • There are areas of increased displacement, where the water waves are higher

• It is possible to analyse superposition clearly when the waves are drawn on a vertical displacement (amplitude)-displacement graph

• Interference is the effect of this superposition

  • Interference can be seen clearly when waves overlap in phase or antiphase 

• When two waves are in antiphase, their combined effect means they cancel each other out

  • This is known as destructive interference

• When two waves are in phase, their combined effect makes the resultant wave amplitude larger

  • This is known as constructive interference

• At points where two waves are neither in phase nor in antiphase, the resultant amplitude is somewhere in between the two extremes

• Individual wave displacements may be positive or negative and are combined in the same way as other vector quantities

Coherence

• For waves to undergo constructive and destructive interference, they must be coherent

• Coherence occurs when waves have:

  • the same frequency

  • a constant phase difference

• Examples of interference from coherent light sources are:  

  • monochromatic laser light 

  • sound waves from two nearby speakers emitting sound of the same frequency

Path difference

• Path difference is defined as:

The difference in distance travelled by the two waves from their sources to the point where they meet

• Path difference vs phase difference:

  • Path difference compares the amount of progress made by waves along a path 

  • Phase difference compares the distance between the phases (peaks and troughs) of coherent waves that are normally travelling parallel to each other at a point 

• The path difference between two coherent waves determines whether there is constructive or destructive interference where they meet

• Path difference is generally expressed in multiples of wavelength

  • Remember to count a whole number of wavelengths (the wave should go up, down, and then back to where it started in one wavelength)

• 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

Conditions for constructive and destructive interference

• In general, for waves emitted by two coherent sources very close together:

  • The condition for constructive interference is:

    • $\mathrm{path}\mathrm{difference}=n\lambda$

  • The condition for destructive interference is:

    • $\mathrm{path}\mathrm{difference}=\left(n+\frac{1}{2}\right)\lambda$

  • Where:

    • $\lambda$ = wavelength of the waves in metres (m)

    • $n$ = 0, 1, 2, 3... (any other integer)

Wavefront diagrams

• Wave behaviour can also be shown by a wavefront diagram

• A curved line represents each wavefront (peak or trough)

• This can show the interference between waves more clearly

• On a wavefront diagram, it is possible to count the number of wavelengths to determine whether constructive or destructive interference occurs at a certain point

• At point P, the number of crests from:

  • Source S₁ = 4λ

  • Source S₂ = 6λ

• So the path difference at P is 6λ – 4λ = 2λ

• This is a whole number of wavelengths, hence, constructive interference occurs at point P