Spatial Dispersion of Acoustic Waves in Functionally Graded Rods

Harmonic acoustic waves in a semi-infinite functional-gradient (FG) one-dimensional rod with arbitrary longitudinal inhomogeneity are analyzed by a combined method based on the modified Cauchy formalism and the method of exponential matrices. Closed dispersion equations for harmonic waves are constructed, from the solution of which implicit dispersion relations for acoustic waves in FG rods are obtained. For longitudinal heterogeneity of polynomial type, the corresponding dispersion relations are constructed explicitly.