Acoustic phonon modes of a free-standing rectangular quantum wire composed of cubic crystals are theoretically investigated using an algorithm developed to analyse data from resonant ultrasound spectroscopy. The normal phonon modes are classified according to their spatial symmetries into a compressional mode termed the dilatational mode and non-compressional modes referred to as the flexural, torsional, and shear modes. The formalism that we present is quite general and can be applied to wires of any cubic material. As an example, the dispersion relations are obtained for square and rectangular wires of GaAs, taking into account the anisotropic elasticity of the material. The dispersion curves for a rectangular wire are compared with those of the approximate hybrid modes referred to as the thickness and width modes, and the validity of the modes is discussed. The existence of edge modes is confirmed by examining the spatial distribution of displacement vectors.
