an elastic wave
An elastic wave is a ripple of deformation traveling through a solid, in which the material's own stiffness provides the restoring force. Tap one end of a steel rail and a disturbance races to the other end; a rock struck in an earthquake sends shudders through the whole planet. In each case bits of material are momentarily displaced and the surrounding elasticity pulls them back, passing the disturbance along -- while the material itself does not travel with the wave.
A bulk isotropic solid carries two distinct kinds of elastic wave, a richer situation than a fluid. Longitudinal (compressional, or P) waves oscillate along the direction of travel, alternately squeezing and stretching the material, and move at c_L = sqrt((lambda + 2 mu)/rho). Transverse (shear, or S) waves oscillate perpendicular to the direction of travel, shearing the material sideways, and move at c_T = sqrt(mu/rho). Because lambda + 2 mu > mu, the P wave is always faster, so it arrives first. Both fall out of the elastodynamic (Navier) equation rho (partial^2 u / partial t^2) = (lambda + mu) grad(div u) + mu laplacian(u).
Elastic waves are how we hear through solids, test materials with ultrasound, and see inside the Earth: seismographs read the P- and S-wave arrivals, and the fact that shear waves cannot pass through the liquid outer core is direct evidence that the core is molten. A key asymmetry: fluids have mu = 0, so they carry only compressional (sound) waves and no shear waves, whereas solids carry both. At the quantum level, the quantized vibrations of a crystal lattice -- these same elastic waves -- are the phonons.
In an earthquake the faster P wave arrives first as a sharp jolt, then the slower, more destructive S wave follows. Timing the gap between them tells a seismologist how far away the quake was -- roughly 8 km for every second of P-minus-S delay in the crust.
Solids carry both faster compressional (P) and slower shear (S) elastic waves; fluids only the former.
The P-before-S rule and the two-speed picture are for waves deep inside a bulk solid. Real solids also carry surface waves (Rayleigh and Love waves) that travel along boundaries with their own speeds and are often the most damaging in earthquakes.