  Phonons in nearly optimally doped HgBa$_2$CuO$_{4+\delta}$ were studied by
inelastic X-ray scattering. The dispersion of the low energy modes is well
described by a shell model, while the Cu-O bond stretching mode at high energy
shows strong softening towards the zone boundary, which deviates strongly from
the model. This seems to be common in the hole-doped high-$T_\mathrm{c}$
superconducting cuprates, and, based on this work, not related to a lattice
distortion specific to each material.
