  Three-dimensional pentamode metamaterials are artificial solids that
approximately behave like liquids, which have vanishing shear modulus.
Pentamodes have recently become experimental reality. Here, we calculate their
phonon band structures for various parameters. Consistent with static continuum
mechanics, we find that compression and shear waves exhibit phase velocities
that can realistically be different by more than one order of magnitude.
Interestingly, we also find frequency intervals with more than two octaves
bandwidth in which pure single-mode behavior is obtained. Herein, exclusively
compression waves exist due to a complete three-dimensional band gap for shear
waves and, hence, no coupling to shear modes is possible. Such single-mode
behavior might, e.g., be interesting for transformation-elastodynamics
architectures.
