  Although metamaterials have been widely used for controlling elastic waves
through bandgap engineering, the directed guidance of stress waves in
non-periodic structures has remained a challenge. This work demonstrates that
spatially graded metamaterials based on conformal mappings present a rich
design space for controlling and attenuating wave motion - without the need for
bandgaps. Conformal mappings transform an elementary unit cell by scaling and
rotation into graded lattices with approximately geometrically similar unit
cells. This self-similarity allows for control over the local wave dispersion
throughout the metamaterial. As a key mechanism, it is shown that elastic waves
cannot propagate through graded unit cells with significant size differences,
except at low frequencies. This is exploited to create low-pass elastic wave
guides, extending beyond classical bandgap engineering, since bandgaps are not
required to achieve wave guiding and attenuation. Experiments confirm the
low-pass elastic wave filtering capability of a planar truss metamaterial with
conformal grading. Finally, a systematic design of curved metamaterial surfaces
is presented, providing a flexible framework for programming low-pass
attenuation and wave guiding in three dimensions.
