Principle of Superposition

• When two or more waves of the same type meet at a point, the resultant displacement is the algebraic sum of the individual displacements

Interference and Coherence

• Interference is the formation of points of cancellation and reinforcement where two coherent waves pass each other.
• Coherence occurs when waves have a constant phase difference.
  

Two-Source Interference

• Conditions for Two-Source Interference:
  • They must meet at a point
  • They must be of the same type
  • They must have the same plane of polarization
• Demonstrating Two-Source Interference:
  • In water:
    • Ripple generators in a tank
  • In light:
    • Double slit interference
  • In microwaves:
    • Two microwave emitters

Formation of Stationary Waves

• A stationary waves is formed when two progressive waves, of the same frequency, amplitude, and speed, travelling in opposite directions are superposed.
• A node is a region of destructive superposition where waves always meet out of phase by π, therefore, displacement = 0.
• An antinode is a region of constructive superposition where waves meet in phase, therefore, the particle vibrates with maximum amplitude.
  
• Neighboring nodes and antinodes are separated by ¹/₂ λ
• Between two adjacent nodes, particles move in phase and they are out of phase with the next two nodes by π.

Stationary Waves at Different Times

Stationary Wave Experiments

• Stretched String
  • A string either attached to a wall or a weight
  • Stationary waves will be produced by the direct and reflected waves in the string.
    

• Microwaves
  • A microwave emitter is placed a distance away form a metal plate that reflects the emitted wave.
  • By moving a detector along the path of the wave, the nodes and antinodes could be detected.
    

• Air Columns
  • A tuning fork held at the mouth of an open tube projects a sound wave into the column of air in the tube.
  • The length can be changed by varying the water level.
  • At certain tube lengths, the air column resonates.
  • This is due to the formation of stationary waves by the incident and reflected sound waves at the water surface.
  • Nodes are always formed at the surface of the water.

Stationary and Progressive Waves

Diffraction

• Diffraction is the spreading of waves as they pass through a narrow slit or near an obstancle.
• For diffraction to occur, the size of the gap should be equal to the wavelength of the wave.
  

Double-Slit Interference

• where:
  • a = split separation
  • D = distance from slit to screen
  • x = fringe width

Diffraction Grating

• where:
  • d = distance between successive slits = reciprocal of the number of lines per meter
  • θ = angle from horizontal equilibrium
  • n = order number
  • λ = wavelength
• Comparing to Double-slit to Diffraction Grating:
  • Maxima are sharper compared to fringes.
  • Maxima are very bright; more slits, more light through.