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Superposition of Waves

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.
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