This paper discusses an undergraduate research project that involves the numerical calculation of phonon dispersion in semiconductor nanostructures. Phonon dispersion is calculated in the elastic continuum approximation by applying a finite-difference method to solving the elastic wave equation with boundary conditions. Students did simulations for two nanostructures: a free-standing GaN thin film and a free-standing AlN/GaN/AlN heterostructure. A comparison of these two structures helps students understand the effect of boundary conditions on the nature of the phonon dispersion. The project is related to current research in phonon thermal transport in semiconductor nanostructures and nanodevices.