Numerical experiments on strong shocks in one-dimensional Toda lattices initially at rest show that the shock is led by a soliton and that the shock versus particle velocity relationship is inconsistent with the Rankine-Hugoniot equations. The shock is at all times unstable in the sense that the shofk structure is a function of the distance of penetration of the shock into the lattice. The wake of strong shocks on the Toda lattice shows an almost monochromatic vibration spectrum, which we take to be the Einstein frequency of the lattice. A conjecture is offered for the reduction of such idealized shock wave data to isothermal equations of state, which bypasses the conventional two-state procedure of application of the Rankine-Hugoniot equations plus the Gr¿neisen equation.