  We study the propagation of an unusual type of periodic travelling waves in
chains of identical beads interacting via Hertz's contact forces. Each bead
periodically undergoes a compression phase followed by a free flight, due to
special properties of Hertzian interactions (fully nonlinear under compression
and vanishing in the absence of contact). We prove the existence of such waves
close to binary oscillations, and numerically continue these solutions when
their wavelength is increased. In the long wave limit, we observe their
convergence towards shock profiles consisting of small compression regions
close to solitary waves, alternating with large domains of free flight where
bead velocities are small. We give formal arguments to justify this asymptotic
behaviour, using a matching technique and previous results concerning solitary
wave solutions. The numerical finding of such waves implies the existence of
compactons, i.e. compactly supported compression waves propagating at a
constant velocity, depending on the amplitude and width of the wave. The beads
are stationary and separated by equal gaps outside the wave, and each bead
reached by the wave is shifted by a finite distance during a finite time
interval. Below a critical wavenumber, we observe fast instabilities of the
periodic travelling waves leading to a disordered regime.
