Acoustic waves of shear horizontal polarization are studied in a finite superlattice (SL) deposited on a substrate with or without a surface cap layer, or sandwiched between two materials. Using a Green's function method, we have obtained closed-form expressions of local and total densities of states as a function of the frequency \ensuremath{\omega} and the wave vector ${\mathit{k}}_{\mathrm{\ensuremath{\parallel}}}$ parallel to the layers. Due to the substrate-SL-capping layer interaction, different kinds of localized and resonant discrete modes are found and their properties are investigated.