The propagation of acoustic or ultrasonic pulses and waves in 1-D media with continuous inhomogeneities due to spatial variations in density, Young modulus, and/or cross section of the propagation medium is discussed. A semianalytical approach leads to a general form of the solution, which can be described by a function, whose Taylor expansion is absolutely convergent. The special case of a periodic inhomogeneity is studied in detail and the dispersion law is found. It is also shown that a finite width pulse is generally not broken down by the inhomogeneity, even though its law of motion is perturbed. A numerical treatment based on the Local Interaction Simulation Approach (LISA) is also considered, and the results of the simulations compared with the semianalytical ones.