Some effects related to high amplitude acoustic waves in periodic media are investigated. The combined action of nonlinearity (harmonic generation) and periodicity (different propagation velocities and attenuation for the different harmonics) results in novel and unexpected phenomena, with respect to the linear problem, and opens the door to new mechanisms of acoustic wave control and manipulation. Sound wave propagation in periodic media has become increasingly popular in the last years, after the introduction of the concept of sonic and phononic crystals. Exploiting the analogies with other type of waves (mainly electrons and light) in the corresponding media, many interesting effects as forbidden propagation bands (band-gaps), focalization, self-collimation, negative refraction, and many others have been proposed. The huge majority of the studies have been done assuming low-amplitude waves, or linear regime, where the mentioned analogies apply. In linear systems, frequency is conserved. Intense wave propagation in nonlinear periodic media has been much less explored, particularly in acoustics. Here we present some of new phenomena related to sound wave propagation in a nonlinear medium with periodically modulated properties. Space dependent linear properties (as density, or sound velocity) introduce dispersion, which may be very strong at some frequency ranges. On the other hand, propagation in a nonlinear medium leads to harmonic generation, which in the case of nondispersive media ends up with the formation of a shock wave. This work explore several effects of strong dispersion in acoustic waves propagating in nonlinear media. The simplest case of nonlinear periodic medium corresponds to a lattice or chain of oscillating masses , coupled by some anharmonic potential and excited at one boundary. Such a lattice, depending on the form of the interation potential, may describe a granular chain (spheres in contact interacting by Hertz potentials), an ionic chain (charges interacting by repulsive Coulomb potentials), and, more generally, a system of masses coupled by nonlinear springs. Most of such complex potentials may be approximated, for small displacements, by a quadratic or cubic dependence on the amplitude, leading to the celebrated FPU lattice. We investigate the process of second harmonic generation in such lattices , under conditions of weak and strong dispersion. Assuming that the wavelength is much larger than the interparticle distance, the lattice behaves as a homogeneous nondispersive medium, where harmonics grow as they propagate propagation; otherwise, dispersion has an impact in the harmonic generation processes. Experiments performed in a chain of repeling magnets are also presented, in good agreement with the main conclusions. Figure 1. Different types of 1D phononic crystals: Left-ionic (top) and granular (bottom) lattices. Right-superlattice.