Waves

1. Transverse: particle motion ⊥ wave direction (e.g. string wave). Needs shear rigidity — travels in solids and along surfaces, not through the bulk of a fluid.
2. Longitudinal: particle motion ∥ wave direction (e.g. sound, as compressions and rarefactions). Needs only bulk elasticity — travels through solids, liquids, and gases.

Wave speed $v$ is the speed of the pattern, not the particle. Transverse particle velocity is $\partial y/\partial t = -A\omega\cos(kx-\omega t+\phi)$ — a different quantity, depends on amplitude and frequency.

T = tension, μ = m/L = linear mass density. This is the wave’s propagation speed, not the transverse particle velocity.

1. Fixed end: reflected pulse inverted (phase change of $\pi$).
2. Free end: reflected pulse upright (no phase change).

Nodes: where $\sin(kx)=0$ (no motion). Antinodes: where $\sin(kx)=\pm1$ (swings through $2A$). Consecutive nodes (or antinodes) are $\lambda/2$ apart; a node sits $\lambda/4$ from the nearest antinode. A standing wave transports no energy — only stores it.

Carrier term oscillates at the average frequency $(f_1+f_2)/2$; the envelope $2A\cos(\pi(f_1-f_2)t)$ swells and fades. Loudness depends on the envelope’s size, which peaks twice per cycle — doubling $(f_1-f_2)/2$ back to $|f_1-f_2|$.

1. Source approaching stationary observer (speed $v_s$): $f_{\text{approach}} = f_0\dfrac{v}{v-v_s}$ — wavefronts crowd closer, frequency rises.
2. Source receding: $f_{\text{recede}} = f_0\dfrac{v}{v+v_s}$ — frequency falls.
3. Observer moving (speed $v_o$) toward/away from a stationary source: $f' = f_0\dfrac{v\pm v_o}{v}$ — wavefront spacing is unchanged; only the interception rate changes.

The speed of sound $v$ is fixed by the medium alone — never affected by source or observer motion.