  We demonstrate that a phonon stopband can be synthesized from an aperiodic
structure comprising a discrete set of phononic filter stages. Each element of
the set has a dispersion relation that defines a complete bandgap when
calculated under a Bloch boundary condition. Hence, the effective stopband
width in an aperiodic phononic filter (PnF) may readily exceed that of a
phononic crystal with a single lattice constant or a coherence scale. With
simulations of multi-moded phononic waveguides, we discuss the effects of
finite geometry and mode-converting junctions on the phonon transmission in
PnFs. The principles described may be utilized to form a wide stopband in
acoustic and surface wave media. Relative to the quantum of thermal conductance
for a uniform mesoscopic beam, a PnF with a stopband covering 1.6-10.4 GHz is
estimated to reduce the thermal conductance by an order of magnitude at 75mK.
