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.
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