  In this work we theoretically study the interface acoustic states of
resonators on a thin plate with topologically protected and conventional
designs. Topologically protected interface state is first analyzed by employing
the conception of breaking inversion symmetry within the unit cell of a
honeycomb lattice for cylindrical and spherical resonators; we further
demonstrate the robustness of the wave propagation along a zig-zag path
containing sharp corners, defect and disorder. The wave propagation ceases to
be preserved if we increase the degree of disorder along the zig-zag path. In
parallel, the conventional interface state is also designed and compared to the
same situations. We found that the conventional interface state suffers back
scattering in the zig-zag path while it can show a more confined wave transport
in some cases. The presence of a defect along the propagation path scatters the
conventional interface wave and in particular can prohibit a full propagation
in presence of a localized state at the defect. If the zig-zag path is made
disordered, the propagation of the conventional interface mode can be conserved
at given frequencies for a low random degree and disappears for higher random
degree as the interface bands become flat in dispersion and turn to localized
states. Finally, we show that the immunity of the topologically protected
design needs the interface to be surrounded by at least two hexagons of the
phononic crystals on both sides, especially at the sharp corners in the zig-zag
path, while the conventional design only needs one hexagon bulk media with the
advantage of compact wave transport. This work puts a step forward for the
interface states in micro-/nano-scale characterization and figures out the
behaviors for both topologically protected and conventional interface states.
