Oscillations and Waves: Harmonic Oscillators, Wave Properties, Interference, Diffraction, Sound
This domain connects the harmonic oscillator to wave phenomena and to the interference and diffraction experiments that dominate the optics and acoustics tasks. Quantitative evaluation of a fringe pattern or a resonance tube is standard.
Harmonic oscillator
A restoring force F = −D·s produces SHM: s(t) = ŝ·sin(ωt + φ) with ω = √(D/m) and T = 2π√(m/D). The pendulum obeys T = 2π√(l/g) only for small amplitudes. Energy alternates between ½Ds² and ½mv²; the total is E = ½Dŝ².
Wave basics
c = λ·f. The medium fixes c, the source fixes f, so λ adjusts when the wave enters another medium. Superposition is linear; only coherent sources with a fixed phase difference give a stable interference pattern.
Interference and diffraction
Double slit (spacing d, screen distance a): path difference Δs = d·sin θ ≈ d·x/a; maxima at d·sin θ = k·λ, fringe spacing Δx = λ·a/d. Single slit of width b: minima at b·sin θ = k·λ. Grating with n lines per metre: d = 1/n, maxima at d·sin θ = k·λ.
Worked example
Double slit with d = 0.25 mm and a = 2.40 m; the third-order maximum (k = 3) lies at x = 1.53 cm. Then λ = x·d/(k·a) = (1.53×10⁻² · 2.5×10⁻⁴)/(3 · 2.40) = 5.3×10⁻⁷ m = 530 nm. Check the small-angle assumption: x/a = 6.4×10⁻³, so replacing sin θ by tan θ changes the result by less than 0.01 %.
Sound
Standing waves in a tube: an open end is an antinode, a closed end a node; a tube closed at one end resonates at L = (2k−1)·λ/4. Two close frequencies produce beats at f_Schwebung = |f₁ − f₂|. An approaching source is heard at a higher frequency (Doppler effect).
Traps
- Counting interference orders from k = 1 at the centre; the centre is k = 0.
- Swapping slit spacing d and slit width b.
- Forgetting to convert lines per millimetre into d in metres.
- Mixing degrees and radians in the calculator.
- Claiming that a louder sound has a higher frequency or a different wavelength in the same medium.