The Lagrangian‐history direct‐interaction approximation for Burgers' equation yields ak−2inertialrange spectrum and an infinite‐Reynolds‐number similarity solution which describes stationary, decaying sawtooth shock waves. This solution differs by a numerical factor from an exact statistical solution of Burgers' equation into which almost all spatially periodic, zero‐mean, infinite‐Reynolds‐number initial ensembles are expected to evolve. The different inertial‐range predictions of the approximation for Burgers' equation and Navier‐Stokes dynamics (wherek−5/3results) are directly associated with the effects of pressure‐induced accelerations on Lagrangian correlation times.