§ 8.1 Stationary waves
Key ideas
- Principle of superposition: where waves overlap, the resultant displacement is the sum of the individual displacements.
- A stationary wave forms when two waves of the same frequency and amplitude travel in opposite directions (typically a wave and its reflection).
- Nodes are points of permanent zero amplitude; antinodes oscillate with maximum amplitude. Adjacent nodes are λ/2 apart.
- A stationary wave stores energy; it does not transfer it along the string or tube.
Fig. 1 · A stationary wave between fixed ends: the solid and dashed envelopes are the string half a period apart; nodes N never move, antinodes A swing fully.
Watch out: adjacent nodes are λ/2 apart, not λ. A string with three nodes spans one whole wavelength, and node-to-antinode is only λ/4.
§ 8.2 Diffraction
Key ideas
- Diffraction is the spreading of a wave as it passes through a gap or around an edge.
- The spreading is greatest when the gap width is comparable to the wavelength.
- Diffraction changes only the direction of spread: frequency, wavelength and speed are unchanged.
Watch out: making the gap narrower spreads the wave more, not less. Diffraction is why you hear (λ ~ 1 m) around a doorway but cannot see (λ ~ 10⁻⁷ m) around it.
§ 8.3 Interference
Key ideas
- Two sources are coherent when they keep a constant phase difference (same frequency); coherence is required for a steady pattern.
- Constructive interference: path difference = nλ. Destructive: path difference = (n + ½)λ.
- In the double-slit experiment, evenly spaced bright and dark fringes appear on the screen; the fringe spacing x grows with wavelength and screen distance, and shrinks as the slits move apart.
Equations
λ = a x / Da = slit separation, x = fringe spacing, D = slit-to-screen distancem
Fig. 2 · Double-slit interference: where the two path lengths differ by a whole number of wavelengths the waves arrive in step, and a bright fringe forms; spacing x = λD/a.
Watch out: x in λ = ax/D is the spacing between adjacent fringes. Measure across several fringes and divide: ten gaps measured once beats one gap measured ten times.
§ 8.4 The diffraction grating
Key ideas
- A grating has thousands of slits, so its maxima are far sharper and brighter than double-slit fringes.
- Orders appear at angles where d sin θ = nλ, for n = 0, 1, 2, ...
- The slit spacing comes from the line density: d = 1 ÷ (lines per metre).
- The highest visible order is capped by sin θ ≤ 1, so n ≤ d/λ.
Equations
d sin θ = n λgrating maxima at order n; d = slit spacingm
Fig. 3 · Grating orders: the straight-through beam is n = 0, with symmetric orders either side at angles given by d sin θ = nλ.
Watch out: "500 lines per mm" means d = 1 ÷ 500 000 m = 2 × 10⁻⁶ m. Convert the line density to per metre before inverting; leaving it per mm inflates every answer a thousandfold.