We consider travelling wave solutions on a one-dimensional lattice, corresponding to mass particles interacting nonlinearly with their nearest neighbour (the 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 given by solutions of a finite-dimensional reversible ordinary differential equation . In particular, near (above) the first critical velocity of the waves, we construct the solitary waves (localized waves with the basic state at infinity) whose global existence was proved by Friesecke and Wattis, using a variational approach. In addition, we find other travelling waves such as (a) a superposition of a periodic oscillation with a non-zero uniform stretching or compression between particles, (b) mainly localized waves which tend towards a uniformly stretched or compressed lattice at infinity, (c) heteroclinic solutions connecting a stretched pattern with a compressed one.