Sound Wave
A sound wave is a longitudinal mechanical wave of compressions and rarefactions that travels through a medium such as air, water, or solids.
Definition
A sound wave is a vibration that travels through air, water, or solid objects to the ear: something shakes, the air around it shakes, and the shaking spreads out until it reaches a listener, but sound needs something to travel through, so in the empty space between planets a rocket engine would be completely silent. It is a longitudinal mechanical wave produced by a vibrating object, traveling as compressions and rarefactions through a medium, with frequency determining pitch and amplitude determining loudness, moving at about $343\ \text{m/s}$ in air at $20^\circ\text{C}$ and faster in liquids and solids. As a pressure wave, $\Delta P = \Delta P_{\max}\sin(kx - \omega t)$, it propagates at $v = \sqrt{B/\rho}$, where the adiabatic bulk modulus for an ideal gas is $B = \gamma P$; intensity is $I = \Delta P_{\max}^2/(2\rho v)$ and falls off as $1/r^2$ from a point source, sound level is $\beta = 10\log_{10}(I/I_0)$ dB, and sound undergoes reflection, refraction, diffraction, interference (beats at $f_{\text{beat}} = |f_1 - f_2|$), and the Doppler effect, with ultrasound above $20\ \text{kHz}$ used for imaging because its short wavelength resolves fine detail.
Example
Plucking a guitar string produces a visible blur that pushes the air back and forth, and those pushes travel across a room to the ear; a bat's $40{,}000\ \text{Hz}$ click has a wavelength in air of $343/40000 \approx 0.0086\ \text{m}$, under a centimeter, small enough to bounce off a moth, with an echo from a moth $2\ \text{m}$ away returning after $4/343 \approx 0.012\ \text{s}$. A firework producing $120\ \text{dB}$ at $10\ \text{m}$ has $I = 10^{12} \times 10^{-12} = 1\ \text{W/m}^2$, dropping to $0.01\ \text{W/m}^2$ (or $100\ \text{dB}$) at $100\ \text{m}$ as intensity falls by $(10/100)^2 = 1/100$, and two flutes at $440\ \text{Hz}$ and $444\ \text{Hz}$ produce $4$ beats per second, the wavering used to tune instruments.
Key Insight
Sound travels about $15$ times faster in steel than in air, so an ear pressed carefully to a railroad track can hear a train long before it arrives through the air. The speed of sound also sets limits in engineering: aircraft crossing $343\ \text{m/s}$ pile up their own pressure waves into a shock cone, and engineers design nozzle shapes, wing sweeps, and even golf-ball dimples around what pressure waves do near that speed.