Summary

A mechanical wave is a disturbance carrying energy through an elastic, inertial medium without net transport of the medium's own matter; in a transverse wave particles vibrate perpendicular to propagation (crests/troughs), in a longitudinal wave they vibrate parallel to it (compressions/rarefactions). A progressive wave is described by $y=A\sin(kx-\omega t)$, with $k=2\pi/\lambda$, $\omega=2\pi f$, and $v=f\lambda=\omega/k$. Wave speed is fixed by the medium: $v=\sqrt{T/\mu}$ for a transverse wave on a stretched string, and for sound in a gas $v=\sqrt{B/\rho}$, where Newton's isothermal assumption ($v_N=\sqrt{P/\rho}$) under-predicted the true speed by about 15-16% until Laplace's adiabatic correction ($v_L=\sqrt{\gamma P/\rho}$) matched experiment. The principle of superposition -- resultant displacement is the sum of individual displacements -- underlies both stationary waves, formed when an incident wave and its own reflection combine into a spatially-fixed pattern of nodes and antinodes ($y=2A\sin(kx)\cos(\omega t)$), and beats, the periodic loudness variation ($f_{\text{beat}}=|f_1-f_2|$) heard when two close frequencies overlap. A string fixed at both ends, and an organ pipe open at both ends, both support the FULL harmonic series, $f_n=nv/2L$; a pipe closed at one end supports only the ODD harmonics, $f_n=(2n-1)v/4L$, with a fundamental $v/4L$ exactly half an open pipe's fundamental $v/2L$ of the same length. The …