Topic 8: Superposition - CAIE Physics A-Level

8.1 Stationary Waves

• Superposition
  • Occurs when two or more waves of the same type cross.
  • Resultant wave displacement is the vector sum of individual wave displacements.
  • Interference
    • Constructive: In-phase waves add displacements, producing larger resultant.
    • Destructive: Out-of-phase waves cancel displacements, producing smaller or zero resultant.
• Stationary Waves
  • Formed by superposition of two progressive waves in opposite directions with same frequency, wavelength, amplitude.
  • No energy transfer occurs.
  • Nodes and Antinodes
    • Antinodes: Maximum displacement (waves meet in phase).
    • Nodes: Minimum displacement (waves meet out of phase).
• Fundamental Frequency
  • Lowest frequency forming a stationary wave (first harmonic).
  • Consists of two nodes and one antinode.
  • Distance between nodes/antinodes is half a wavelength.
• Harmonics
  • Second harmonic: Double the fundamental frequency (two antinodes).
  • Third harmonic: Triple the fundamental frequency (three antinodes).

Examples of Stationary Waves

• Microwaves: Reflect beam at metal plate, probe to find nodes and antinodes.
• Sound Waves: Speaker at one end of closed tube, powder moves at antinodes, settles at nodes.

8.2 Diffraction

• Definition: Spreading of waves through/around a gap.
• Greatest when gap size equals wavelength.
• Single Slit Diffraction
  • Monochromatic light diffracted through a slit forms light/dark fringe pattern.
  • Bright central fringe (constructive interference), alternating dark/bright fringes.

8.3 Interference

• Coherent Light Source
  • Same frequency and wavelength, fixed phase difference (e.g., lasers).
• Young's Double Slit Experiment
  • Two coherent sources or one source with double slit to form interference pattern.
  • Light through slits diffracts, forming light (constructive) and dark (destructive) fringes.
  • Formula: ( x = \frac{\lambda D}{a} )
    • ( x ): Fringe spacing; ( \lambda ): Wavelength; ( D ): Distance to screen; ( a ): Slit separation.

8.4 The Diffraction Grating

• Definition: Slide with many close, equal-spacing slits.
• Pattern: Sharper, brighter than double slits; multiple rays reinforce pattern.
• Order Lines
  • Zero order line: Central bright fringe.
  • First, second order lines: Constructive interference on each side.
• Formula: ( \sin \theta = \frac{n\lambda}{d} )
  • ( d ): Slit distance; ( \theta ): Angle to normal; ( n ): Order; ( \lambda ): Wavelength.
  • ( \sin \theta \leq 1 ) determines possible values.
• Applications
  • Analyzing star light spectra for elements.
  • X-ray crystallography for atomic spacing.