Derive the equation of a simple harmonic wave travelling along the positive x direction. Hence show that the wave speed is v = ω/k = νλ.

1. 1 · Start with the source. The particle at O vibrates as y = a sin ωt, where a is the amplitude and ω = 2πν.
2. 2 · Find the phase lag at P. A point P at distance x lags behind by φ. A path of λ means a phase of 2π, so φ = (2π/λ)x = kx.
3. 3 · Draw the wave. [Sine curve along OX; mark O, a point P at distance x, the amplitude a, the wavelength λ and the direction of travel]
4. 4 · Write the equation. y = a sin(ωt − kx) = a sin 2π(t/T − x/λ).
5. 5 · Find the wave speed.
   • The wave moves λ in one period T, so v = λ/T = νλ
   • ω/k = 2πν ÷ (2π/λ) = νλ, so v = ω/k
6. 6 · Close in one line. So, a harmonic wave is y = a sin(ωt − kx), and it travels at v = ω/k = νλ.

Scoring tip: Write what each symbol means (a, ω, k, x, t). Examiners give a mark for the meanings.