We investigate the vibration properties of adsorbed nanostructure on the infinite square crystalline surface. The surface is considered as an infinite slab of one atomic layer, and the nanostructure as an isolated diatomic molecule chain on the surface of a cubic lattice which is parallel to y-axis, and takes three different positions: top, hollow and bridge. The vibrational dynamics of the structure is considered within the harmonic approximation framework. The evanescent and propagating vibrational field of the perfect lattice is determined and discussed. The presence of the molecule chain breaks down the translation symmetry in one direction and gives rise to localized states on its neighborhood. The mathematical framework of the matching method is used to analyze the localization phenomena at the nanostructure boundaries. Typical dispersion curves for modes of energies along the inhomogeneity are given with their polarizations. The fine structure of the spectrum and its origins are clearly identifiable, which gives a new insight into the localization problem. Furthermore, the existence and nature of the localized phonons like modes associated with an isolated defect are derived, and the importance of the contribution of these modes to the spectral and states densities is exhibited clearly.