Over 200 crossings of the distant downwind flanks of Earth's magnetosonic bow shock by ISEE 3 included many cases of weak, or low Mach number, quasiparallel shocks. A consistent feature of the magnetic field profiles was the presence of large amplitude, near periodic to irregular transverse oscillations downstream from even the weakest Q∥ shocks. Large downstream perturbations with whistler-like features similar to those of the observations appear in one-dimensional simulations when MA>2.5 but not when MA=2.1. The observed cases with downstream waves also occurred when MA>≈2.5, suggesting the importance of the Alfv¿n as opposed to magnetosonic Mach number in determining the signature of weak Q∥ shocks. ¿American Geophysical Union 1991