Overview
This lecture covers wave superposition, stationary waves, diffraction, interference, and diffraction gratings, detailing concepts, equations, and key experimental applications for each.
Superposition of Waves
- Superposition is when wave displacements combine at a point to produce a resultant displacement.
- Constructive superposition happens when waves are in phase (phase difference of 0, 2π, 4π...), resulting in maximum amplitude.
- Destructive superposition occurs when waves are in antiphase (phase difference π, 3π, 5π...), resulting in minimum or zero amplitude.
Stationary (Standing) Waves
- Stationary waves are formed by the superposition of two identical waves traveling in opposite directions.
- Nodes occur at points of destructive interference (zero amplitude); antinodes at constructive interference (maximum amplitude).
- On a string of length L: fundamental wavelength λ = 2L, first harmonic f = v/2L.
- For higher harmonics: nth harmonic wavelength λ = 2L/n, frequency f_n = n·(v/2L).
- In pipes: closed-end pipes have fundamental λ = 4L, f = v/4L; open-end pipes have fundamental λ = 2L, f = v/2L.
Diffraction
- Diffraction is the spreading of waves when they pass through a gap or around an obstacle.
- Maximum diffraction occurs when gap/obstacle size ≈ wavelength.
- For light, significant diffraction at slits < 0.1 mm; longer wavelengths diffract more than shorter ones.
Interference of Waves
- Interference is the pattern from two coherent waves overlapping, forming nodes (destructive) and antinodes (constructive).
- Coherence requires constant phase difference and same frequency/wavelength.
- Path difference for nodes: (n+½)λ; for antinodes: nλ.
- In Young's double slit: fringe spacing x = λD/a (λ=wavelength, D=distance to screen, a=slit separation).
Diffraction Gratings
- A diffraction grating contains many closely spaced slits producing sharp, bright maxima.
- Maxima occur at angles θ where d·sinθ = nλ (d=grating spacing, n=order).
- Number of observable maxima depends on grating lines per mm and wavelength.
- Grating spacing d = 1/number of lines per unit length.
Key Terms & Definitions
- Superposition — Combination of displacements from two or more waves at a point.
- Node — Point on a stationary wave with zero amplitude.
- Antinode — Point on a stationary wave with maximum amplitude.
- Coherent Waves — Waves with constant phase difference and same frequency.
- Diffraction — Spreading of waves past obstacles or through gaps.
- Diffraction Grating — Optical device with many slits for creating interference patterns.
Action Items / Next Steps
- Practice calculating fundamental and harmonic frequencies for strings and air columns.
- Solve problems using double slit and diffraction grating equations.
- Review definitions and distinguish between constructive and destructive interference.