Oscillations and waves: harmonic oscillator, wave properties, interference, diffraction, superposition
What this domain tests
Oscillations and waves covers the harmonic oscillator, the description of waves, superposition, interference and diffraction. Tasks connect a pendulum or spring experiment with theory, read fringe patterns, and ask you to explain why a pattern appears rather than only computing a value.
Harmonic oscillator
- Restoring force F = −D·s, so a = −ω²s with ω = 2π/T.
- Spring: T = 2π√(m/D); pendulum for small angles: T = 2π√(l/g).
- Energy oscillates between ½Ds² and ½mv²; the total is E = ½D·ŝ².
- Damping reduces the amplitude; resonance occurs when the driving frequency matches the natural frequency.
Waves
- c = λ·f = λ/T; transverse waves oscillate perpendicular to the propagation direction, longitudinal waves along it.
- Superposition adds displacements; only coherent sources with a constant phase difference produce a stable interference pattern.
- Path difference: constructive Δs = kλ, destructive Δs = (k + ½)λ.
- Double slit: d·sin θ_k = kλ; a grating uses the same condition with sharper maxima. Single slit: a·sin θ_k = kλ gives the minima.
Worked example
A double slit with d = 0.50 mm stands 2.0 m from a screen and is illuminated with λ = 633 nm. The fringe spacing is Δx = λ·L/d = (633·10⁻⁹ m · 2.0 m)/(5.0·10⁻⁴ m) = 2.5·10⁻³ m, so consecutive maxima are 2.5 mm apart. Cross-check with d·sin θ = λ: sin θ = 1.27·10⁻³, and for small angles tan θ ≈ sin θ, so x = L·tan θ ≈ 2.5 mm - the same result. Always name the condition you use and check whether the small-angle approximation is justified.
Traps
- Mixing the single-slit minimum condition with the double-slit maximum condition.
- Using degrees in a formula derived for radians, or skipping the small-angle check.
- Assuming any two sources interfere - coherence is required.
- Confusing period T, frequency f and angular frequency ω.
- Forgetting that in a standing wave neighbouring nodes are λ/2 apart, not λ.