Theoretical and numerical analyses of acoustical waves propagating in a functionally graded (FG) rod with exponential inhomogeneity and satisfying the generalized Wiechert condition, reveals (i) a constant propagation velocity, which depends upon Wiechert parameter; (ii) varying acoustic impedance; (iii) varying stress amplitudes; and, (iv) varying particle velocities, implying the difference in the direction of propagation (chirality with respect to stress and particle velocity amplitudes). The analyses relate to the propagation of delta-like stress pulses.