Transverse Wave
A transverse wave makes the medium vibrate perpendicular to the direction the wave travels, forming crests and troughs like a rope wave.
Definition
A transverse wave wiggles perpendicular to the direction it travels, so the particles of the medium move up and down (or side to side) while the wave itself moves sideways, forming crests at the highest points and troughs at the lowest, with the distance from crest to crest equal to one wavelength. Waves on strings, ripples on water surfaces, and all electromagnetic waves, including light and radio, are transverse. Mathematically, displacement is perpendicular to the propagation vector, $y(x,t) = A\sin(kx - \omega t)$ with $y \perp x$, and the transverse velocity of a medium element, $\partial y/\partial t = -A\omega\cos(kx - \omega t)$, is distinct from the wave speed $v = \omega/k$; because transverse mechanical waves require a shear restoring force, they propagate in solids and along interfaces but not through the bulk of fluids, and electromagnetic waves are transverse with $\vec{E} \perp \vec{B} \perp \vec{k}$, giving rise to polarization.
Example
Shake one end of a jump rope up and down and humps travel along it toward a friend even though the rope itself only moves up and down; shaking it $3$ times per second with crests $0.5\ \text{m}$ apart gives a wave speed of $v = f\lambda = 3 \times 0.5 = 1.5\ \text{m/s}$. A string wave with $A = 0.02\ \text{m}$, $f = 50\ \text{Hz}$, and $v = 20\ \text{m/s}$ has wavelength $0.4\ \text{m}$ and a maximum transverse point speed of $A\omega = 0.02 \times 2\pi \times 50 \approx 6.3\ \text{m/s}$, even though the pattern itself travels at $20\ \text{m/s}$ sideways.
Key Insight
Light is a transverse wave, which is why it can be polarized, and polarized sunglasses block light waves vibrating horizontally to cut glare from roads and water. Seismic S-waves (transverse) vanish in Earth's shadow zone while P-waves (longitudinal) pass through, and that single observation revealed the planet's liquid outer core without anyone drilling more than a few kilometers.