  Dynamical properties of homogeneous Fermi-Fermi mixtures of dipolar and
non-dipolar atoms are studied at zero temperature, where dipoles are polarized
by an external field. We calculate the density-density correlation functions in
a ring-diagram approximation and analyze the pole structure to obtain
eigenfrequencies of collective excitations. We first determine stability phase
diagrams for the mixtures available in experiments: $^{167}$Er-$^{173}$Yb,
$^{167}$Er-$^{6}$Li, $^{161}$Dy-$^{173}$Yb, and $^{161}$Dy-$^{6}$Li systems,
and show that the mixtures with larger mass imbalance tend to be more unstable.
We then investigate the parameter dependence of an undamped zero sound with an
anisotropic real dispersion relation in the stable phase for the
$^{161}$Dy-$^{173}$Yb mixture, and the speed of sound exhibits a critical angle
of possible propagation with respect to the dipole polarization direction,
above which the sound mode disappears in the particle-hole continuum. Since the
sound mode is a coherent superposition of density fluctuations of dipolar and
non-dipolar atoms, the existence of the sound mode, e.g., the value of the
critical angle, is significantly affected by the inter-particle interaction
through the density-density correlation between dipolar and non-dipolar atoms.
We have also observed such an effect of the inter-particle interaction in the
study of a linear response of density fluctuations to an external perturbation.
