  We present examples of body wave and surface wave propagation in deformed
solids where the slowest and the fastest waves do not travel along the
directions of least and greatest stretch, respectively. These results run
counter to commonly accepted theory, practice, and implementation of the
principles of acousto-elasticity in initially isotropic solids. For instance we
find that in nickel and steel, the fastest waves are along the direction of
greatest compression, not greatest extension (and vice-versa for the slowest
waves), as soon as those solids are deformed. Further, we find that when some
materials are subject to a small-but-finite deformations, other extrema of wave
speeds appear in non-principal directions. Examples include nickel, steel,
polystyrene, and a certain hydrogel. The existence of these ``oblique'',
non-principal extremal waves complicates the protocols for the non-destructive
determination of the directions of extreme strains.
