  The explanation of the basic acoustic properties of crystals requires a
recognition of the acoustic axes. To derive the acoustic axes in a given
material, one requires both a workable method and the necessary and sufficient
criteria for the existence of the acoustic axes in a partial propagation
direction. We apply the reduced form of the acoustic tensor to the acoustic
axis conditions in the present work. Using this tensor, we obtain in a compact
form, allowing for qualitative analysis, the necessary and sufficient criteria
for the existence of the acoustic axis. Furthermore, the well-known Khatkevich
criteria and their variants are recast in terms of the reduced acoustic tensor.
This paper's primary input is an alternate minimal polynomial-based system of
acoustic axes conditions. In this approach, we derive an additional
characteristic of acoustic axes: the directions in which the minimal polynomial
of the third order is reduced to that of the second order. Next, we offer a
general solution to the second-order minimum polynomial equation, that utilizes
a scalar and a unit vector for defining the acoustic tensor along the acoustic
axis. It is shown that the scalar matches the eigenvalue of the reduced
acoustic tensor, and the vector corresponds to the polarization into the single
eigenvalue direction. We use the minimal polynomial construction to demonstrate
the equivalence of different acoustic axis criteria. We demonstrate the
applicability of this approach to actual computations of the acoustic axes and
their fundamental properties (phase speeds and polarizations) for high symmetry
cases, such as isotropic materials and RTHC crystals.
