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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.

###### 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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