The authors study the asymptotic behaviour of the infinite Toda lattice subject to a step-like initial condition by analyzing the Whitham (averaged) equations for this system. They are able to characterize the essential structure of the solution in the presence of shocks that are due to the slow modulations of multiphase wave trains for the Toda lattice. This problem was studied earlier [see B. L. Hollan et al., Shock waves in the Toda lattice: Analysis, Phys. Rev. A 24, 2595-2623 (1981)] from a numerical point of view. The predicted qualitative behaviour of the solutions in this paper agrees with the numerical results.