IBDP Physics Unit 9

Wave phenomenon, SHM, interference, slits , doppler effects and resolution

U9 Wave Phenomena

• 9.1 Simple Harmonic Motion
  • Definition
    • Simple harmonic motion (SHM) is motion about a fixed equilibrium position where the acceleration of the body is proportional to its displacement, but in the opposite direction.
    • • F = -kx
    • ω = 2π/T = 2πf
  • Formulas
    • a = -kx => a = −ω^2*x
      • a = −ω^2*x => ω^2 = k/m => ω = √(k/m)
    • • x∝sin(ωt) => x=(x0)*sin(ωt) => v=ω*(x0)*cos(ωt)
      • v = ±ω*√(x0^2 - x^2)
      • sin θ ≈ θ
    • T(spring) = 2π√(m/k); T(pendulum) = 2π√(l/g)
    • Ek = 1/2*k*x^2
• 9.2 Single Slit Diffraction
  • Definition
    • The pattern that we see when waves diffract through a single slit is called the single-slit diffraction pattern.
    • Single-slit interference
      • The central maximum is larger than the slit width.
      • The central maximum is very much more intense than the other maxima
      • Each successive maximum is just less than half the intensity of the previous maximum.
      • The subsidiary maxima decrease in intensity the further out they get.
      • The central maximum is twice the width of the other maxima, which are all evenly spaced.
    • Occur because of the interference of the waves coming from the top and bottom of the slit. Using Huygens' principle we know that there are wave sources all along the slit. To find the first minimum, we must find two waves that have a path difference equal to λ/2 to create destructive interference.
  • Changing Slit Width
  • Formulas
    • • θ=λ/b
      • λ/2 = (b/2)*sinθ
• 9.5 Doppler Effect
  • Definition
    • Essential idea: The Doppler effect describes the phenomenon of wavelength frequency shift when relative motion occurs.
  • Formula
    • f′ = f(v/(v±v_s))
      • λ′ = (v±v_s)/f
        • － for towards observer
        • + for moving away
    • Δf/f = Δλ/λ ≈ v/c
• 9.4 Resolution
  • Definition
    • Two diffraction patterns very close together. It is easy to see that if the peaks were wider the two images would blur together.
    • Rayleigh's criterion
      • Definition: The Rayleigh criterion states that two sources can resolve if the principal maximum of one source is no closer than the first minimum of the other.
      • On the left, the images are easily resolved: The blue principal maximum is further away than the first minimum of the red principal maximum. The middle diagram shows the critical case: The red principal maximum is exactly in line with the first blue minimum.
      • A full mathematical description of a circular aperture tells us that we need to modify the single slit equation by a factor of 1.22. This derivation is beyond the scope of this course, but it tells us that for two images to be just resolved: θ≥1.22λb
• 9.3 Interference
  • Double-slit
    • Young's double-slit experiment
      • Experiment setup
      • The 'bright fringe' region indicates where constructive interference is found: where waves meet with a path difference = nλ, where n=0,±1,±2,...
      • Regions labelled ‘dark fringe’ indicates where destructive interference takes place: where waves meet with a path difference = (n+1/2)λ, where n=0,±1,±2,...
    • Coherent
      • Therefore, the double-slit can cause both interference and coherence
  • Multiple slits diffraction
    • Double Slits
      • Definition
        • When two slits are used to diffract a source each slit produces a diffraction pattern and interference occurs.
      • 1. Intense, dominant central maximum. 2. Less intense subsequent maxima. 3. The position of these maxima is in the same position as the single-slit pattern.
    • Multiple Slits
      • • d sinθ = nλ
      • Diffraction gratings often quote ‘500 lines per mm’. To use this information in the diffraction grating equation, you have to convert this data into the distance between the slits, d. d = 1mm/500 = 2×10−6m
  • Formulas
    • s = λD/d
    • • d sinθ = nλ
      • The two rays of light (red) meet at the first maximum away from the centre. Their path difference is λ. It is important to remember that D>>d, which means we can approximate both of the marked right angles to exactly 90∘ and say that the two angles marked θ (green and blue) are equal.