  This work investigates the echoes in axial gravitational perturbations in
compact objects. To this end, we propose an alternative scheme of the finite
difference method implemented in two coordinate systems, where the initial
conditions are placed on the axis of the tortoise coordinate with appropriate
boundary conditions that fully respect the causality. The scheme is then
employed to study the temporal profiles of the quasinormal oscillations in the
Schwarzschild black hole and the uniform-density stars. When presented in a
two-dimensional evolution profile, the resulting ringdown waveforms in the
black hole metric are split into the reflected and transmitted waves as the
initial perturbations evolve and collide with the peak of the effective
potential. On the other hand, for compact stars, quasinormal oscillations might
be characterized by echoes. Consistent with the causality arguments, the
phenomenon is produced by the gravitational waves bouncing between the
divergent potential at the star's center and the peak of the effective
potential. The implications of the present study are also discussed.
