We consider travelling wave solutions on a one-dimensional lattice, corresponding to mass particles interacting nonlinearly with their nearest neighbor (Fermi-Pasta-Ulam model). A constructive method is given, for obtaining all small bounded travelling waves for generic potentials, near the first critical value of the velocity. They all are solutions of a finite dimensional reversible ODE. In particular, near (above) the first critical velocity of the waves, we construct the solitary waves whose global existence was proved by Friesecke et Wattis [1], using a variational approach. In addition, we find other travelling waves like (i) superposition of a periodic oscillation with a non zero averaged stretching or compression between particules, (ii) mainly localized waves which tend to uniformly stretched or compressed lattice at infinity, (iii) heteroclinic solutions connecting a stretched pattern with a compressed one.