  Inelastic neutron scattering has been used to study the magneto-elastic
excitations in the multiferroic manganite hexagonal YMnO$_3$. An avoided
crossing is found between magnon and phonon modes close to the Brillouin zone
boundary in the $(a,b)$-plane. Neutron polarization analysis reveals that this
mode has mixed magnon-phonon character. An external magnetic field along the
$c$-axis is observed to cause a linear field-induced splitting of one of the
spin wave branches. A theoretical description is performed, using a Heisenberg
model of localized spins, acoustic phonon modes and a magneto-elastic coupling
via the single-ion magnetostriction. The model quantitatively reproduces the
dispersion and intensities of all modes in the full Brillouin zone, describes
the observed magnon-phonon hybridized modes, and quantifies the magneto-elastic
coupling. The combined information, including the field-induced magnon
splitting, allows us to exclude several of the earlier proposed models and
point to the correct magnetic ground state symmetry, and provides an effective
dynamic model relevant for the multiferroic hexagonal manganites.
