  The phononic band structures of two-dimensional solid phononic crystals with
different lattice and scatterer symmetry are studied numerically, with three
types of lattice (square, triangular and rectangular) and four different
scatterer shapes (circle, hexagon, square and rectangle) considered. XY and $Z$
vibration modes are investigated separately. Two types of phononic crystal are
considered: one composed of high-density rods embedded in a low-density matrix,
the other of low-density rods in a high-density matrix. In the former case,
lattice type and polarization being fixed, the broadest gaps are obtained when
the symmetry of the rods corresponds to that of the lattice (the shape of a rod
is identical with that of the first Brillouin zone); the largest gap width
values are observed in triangular lattice-based crystals (compared to those
based on square and rectangular lattices), the shape of the corresponding first
Brillouin zone being closest to a circle. These rules do not apply to
structures in which the density of the rod material is lower than that of the
matrix. In this case, when the symmetry of the rods corresponds to that of the
lattice, gaps either fail to appear at all, or are much narrower than in other
configurations. The effect of other material parameter values (such as the
longitudinal and transversal velocity values) on the considered relation proves
much lesser.
