Interference of Waves (HL IB Physics)

• Double-source interference involves producing a diffraction and an interference pattern using either:

• • The interference of two coherent wave sources 
  • A single wave source passing through a double slit
• Examples of double-source interference include:
  • A laser beam that creates bright and dark fringes on a screen
  • Two speakers emitting a coherent sound
  • Microwaves diffracted through two slits

Sound wave interference from two speakers emitting a coherent sound

A microwave interference experiment creates a diffraction pattern with regions where microwaves are and are not detected

Interference

• Interference is the effect observed due to the superposition of two or more waves
  • It can be seen clearly when waves overlap completely in phase or antiphase

• The maximum amount of superposition occurs when two waves are in phase

• • They meet either peak-to-peak or trough-to-trough
  • This results in the two waves adding together
  • This is called constructive interference

• The minimum amount of superposition occurs when two waves are in antiphase

• • They meet peak-to-trough
  • This results in the two waves cancelling each other out and having zero effect (there is an effect - that they cancel out)
  • This is called destructive interference

• Constructive and destructive interference occurs when waves are coherent

Waves undergo the maximum amount of constructive and destructive interference when they are in phase or antiphase

Coherence

• For waves to be coherent they must have:
  • The same frequency
  • A constant phase difference

Coherent vs. non-coherent waves. The abrupt change in phase creates an inconsistent phase difference.

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

• Examples of interference from coherent light sources are:  
  • Monochromatic laser light 
  • Sound waves from two nearby speakers emitting sound of the same frequency

• Whether waves are in phase or antiphase is determined by their path difference

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 vs phase difference
  • Phase difference compares the distance between the phases (peaks and troughs) of waves that are normally travelling parallel to each other at a point
  • Path difference compares the amount of progress made by waves along a path, so the difference in the distance travelled by the two waves

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) =  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 = 

• The condition for destructive interference is:

path difference =

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

Path Difference and Wavefront Diagrams

• Wavefront diagrams show the interference between waves more clearly

At the blue dot where the peak of two waves meet, constructive interference occurs. At the yellow dot where two troughs meet, constructive interference also occurs. Constructive interference occurs along the lines of maximum displacement. At the green dot, where a peak and a trough meet, destructive interference occurs

• 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 waves have a path difference of a whole number of wavelengths, resulting in constructive interference

• 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