where
v$_{l}$ is the longitudinal sound velocity, and
v$_{tl}$ and
v$_{t2}$ are the transverse sound velocities in the first and second mode, respectively. As seen in these two formulas above,
C$_{33}$ and
C$_{44}$ determine the longitudinal and transverse sound velocities along the [
001] propagation direction, respectively, and
C$_{11}$ and
C$_{44}$ dominate sound velocities along the [
100] propagation direction. The calculated anisotropic sound velocities of C$_{3}$, Si$_{3}$, and Ge$_{3}$ in
P6$_{2}$22 phase are listed in
Table 5. It can be seen that the highest sound velocity among these three Group IV element allotropes is 17843 m/s appeared in
P6$_{2}$22-C$_{3}$, and the lowest sound velocity is 2900 m/s appeared in
P6$_{2}$22-Ge$_{3}$, both of which are exhibited in the [
100] propagation direction. In the [
001] propagation direction, each of these three crystals demonstrates the symmetrical transverse sound velocity in the first and second mode. Meanwhile, it is obvious that the highest sound velocity along different directions is the longitudinal sound velocity in the [
001] propagation direction, and the transverse sound velocity in the first mode in the [
100] propagation direction. In addition, since the density of
P6$_{2}$22-C$_{3}$ (5.647 g/cm$^{3}$) is the highest while its elastic constants (see
Table 2) are the smallest among these three crystals, its sound velocities along different directions are certainly lower than those of
P6$_{2}$22-C$_{3}$ and Si$_{3}$ as shown in
Table 5. The sound velocities along different directions of
P6$_{2}$22-C$_{3}$ are all largely higher than those of
P6$_{2}$22-Si$_{3}$, which is due to the fact that the dominant elastic constants (
C$_{11}$,
C$_{33}$, and
C$_{44}$) of
P6$_{2}$22-C$_{3}$ largely outweigh those of
P6$_{2}$22-Si$_{3}$, even though the density of
P6$_{2}$22-Si$_{3}$ (2.574 g/cm$^{3}$) smaller than that of
P6$_{2}$22-C$_{3}$ (3.647 g/cm$^{3}$).