  We report on first investigation of the lattice dynamics in the novel
superconducting material Na$\_{0.35}$CoO$\_2$$\cdot$1.3H$\_2$O and the
non-hydrated parent compound Na$\_{0.7}$CoO$\_2$ by inelastic x-ray scattering.
The measured phonon dispersion along the $\Gamma-M$ direction show a marked
softening with hole doping of two optical phonon branches close to the
Brillouin zone boundary. The phonon spectra, dispersion, and softening are well
reproduced by first-principle calculations. The calculations indicates that the
soft branches are mainly composed of Co-vibration modes. The estimation of the
critical temperature based on electron-phonon coupling mechanism undisputedly
points to a non-conventional superconducting state in this material.
