  Magnons in ferromagnets have one chirality, and typically are in the GHz
range and have a quadratic dispersion near the zero wavevector. In contrast,
magnons in antiferromagnets are commonly considered to have bands with both
chiralities that are degenerate across the entire Brillouin zone, and to be in
the THz range and to have a linear dispersion near the center of the Brillouin
zone. Here we theoretically demonstrate a new class of magnons on a
prototypical $d$-wave altermagnet RuO$_2$ with the compensated antiparallel
magnetic order in the ground state. Based on density-functional-theory
calculations we observe that the THz-range magnon bands in RuO$_2$ have an
alternating chirality splitting, similar to the alternating spin splitting of
the electronic bands, and a linear magnon dispersion near the zero wavevector.
We also show that, overall, the Landau damping of this metallic altermagnet is
suppressed due to the spin-split electronic structure, as compared to an
artificial antiferromagnetic phase of the same RuO$_2$ crystal with
spin-degenerate electronic bands and chirality-degenerate magnon bands.
