Using the finite difference time domain method, we investigate theoretically the band structure and phonon transport in a
phononic crystal constituted by a periodical array of cylindrical dots deposited on a thin plate of a homogeneous
material. We show that this structure can display a low frequency gap, as compared to the acoustic wavelengths in the
constituent materials, similarly to the case of locally resonant structures. The opening of this gap is discussed as a
function of the geometrical parameters of the structure, in particular the thickness of the homogeneous plate and the
height of the dots. We show the persistence of this gap for various combinations of the materials constituting the plate
and the dots. Besides, the band structure can exhibit one or more higher gaps whose number increases with the height of
the cylinders. The results are discussed for different shapes of the cylinders such as circular, square or rotated square.
The band structure can also display an isolated branch with a negative slope which can be useful for the purpose of
negative refraction phenomena. We discuss the condition to realize wave guiding through different types of linear
defects inside such a phononic crystal. Finally, we investigate the phonon transport between two substrates connected by
a periodic array of dots. We discuss different features appearing in the transmission spectra such as the Perot-Fabry type
oscillations related to the height and the nature of the dots, the existence of transmission gaps related to the period and
the nature of the substrate, the possibility of a narrow transmission peak close to a zero of transmission.