The whole chapter, on one bench

Simple harmonic motion — acceleration is proportional to displacement and always directed toward the mean position: a = −ω²x. The projection of uniform circular motion onto a diameter (the phasor / reference circle).

Amplitude A — greatest displacement. Phase constant φ — where in the cycle the motion starts. Velocity leads displacement by π/2; acceleration by π.

Springs — in parallel the effective constant adds (k = k₁+k₂, stiffer); in series the compliances add (1/k = 1/k₁+1/k₂, softer). Damped: amplitude decays as e^−bt/2m. Resonance: driving at the natural frequency ω₀ gives the largest response.

Progressive wave — a disturbance that carries energy and momentum, not matter. Transverse: particles move ⊥ to travel (waves on a string, light). Longitudinal: particles move along travel (sound).

Particle velocity u = ∂y/∂t is the slope of the displacement–time graph; the wave velocity v is fixed by the medium. They are linked by u = −v·(∂y/∂x): a particle's speed is the wave speed times the local slope.

Power & intensity — a string carries P = ½μω²A²v; intensity ∝ A². Reflection: at a fixed end the pulse flips (π phase change); at a free end it does not.

Superposition — where waves overlap, displacements add. Two coherent waves give interference: in phase (δ = 2nπ) build up, out of phase (δ = (2n−1)π) cancel.

Standing wave — two equal waves travelling opposite ways: y = 2A sin(kx) cos(ωt). Fixed nodes (always still) and antinodes (largest swing); no net energy transport. Node-to-node = λ/2, node-to-antinode = λ/4.

Strings & pipes — boundary conditions pick discrete frequencies (harmonics). An open pipe gives all harmonics; a closed pipe only odd ones. End correction e ≈ 0.6r extends the effective length. Melde's experiment relates loop count to tension (p ∝ 1/√T).

Speed of sound — Newton assumed isothermal (v = √P/ρ, too low); Laplace corrected it to adiabatic (v = √γP/ρ). It rises with temperature (v ∝ √T) and, in air, with humidity.

Displacement vs pressure wave — the pressure wave leads the displacement wave by π/2: where displacement is zero (a node of s) the pressure swing is greatest. Amplitude p₀ = Bk·s₀ = ρvω·s₀.

Intensity & loudness — intensity falls as 1/r² from a point source; loudness in decibels is β = 10 log(I/I₀). Resonance tube finds the speed of sound; Quincke's tube shows interference by a sliding path difference.

Beats — two tones of nearly equal frequency alternately reinforce and cancel, giving a slow throb in loudness. The result is a fast carrier at the average pitch inside a slow amplitude envelope.

Beat frequency equals the difference of the two frequencies (one loud–soft cycle per hertz of difference). Careful: the envelope repeats twice per carrier-beat, so the audible beat rate is Δf, not Δf/2. Wax on a fork adds mass and lowers its frequency — the trick to tell which of two forks is the faster one.

Doppler effect — relative motion between source and observer changes the observed frequency. Approach bunches the wavefronts (higher pitch); recession stretches them (lower pitch). Only the component along the line joining them counts.

Sign rule — in f′ = f₀(v ± v_o)/(v ∓ v_s), pick the sign that raises the pitch for approach. Speeds are measured relative to the medium (the air).

Wind & reflection — a wind changes the ground-frame sound speed to v + w (it shifts nothing for a stationary source and observer). A signal reflected off a moving target Dopplers twice — the basis of radar and sonar speed guns. Sonic boom: v_s = v makes the formula diverge.