  Black hole perturbation theory for Kerr black holes is best studied in the
Newman Penrose Formalism, in which gravitational waves are described as
perturbations in the Weyl scalars $\psi_0$ and $\psi_4$, with the governing
equation being the well-known Teukolsky equation. Near infinity and near
horizon, $\psi_4$ is dominated by the component that corresponds to waves
propagating towards the positive radial direction, while $\psi_0$ is dominated
by the component that corresponds to waves that propagate towards the negative
radial direction. Since gravitational-wave detectors measure out-going waves at
infinity, research has been mainly focused on $\psi_4$, leaving $\psi_0$ less
studied. But the scenario is reversed in the near horizon region where the
in-going-wave boundary condition needs to be imposed. Thus, the near horizon
phenomena, e.g., tidal heating and gravitational-wave echoes from Extremely
Compact Objects (ECOs), require computing $\psi_0$. In this work, we explicitly
calculate the source term for the $\psi_0$ Teukolsky equation due to a point
particle plunging into a Kerr black hole. We highlight the need to regularize
the solution of the $\psi_0$ Teukolsky equation obtained using Green's function
techniques. We suggest a regularization scheme for this purpose and go on to
compute the $\psi_0$ waveform close to a Schwarzschild horizon for two types of
trajectories of the in-falling particle. We compare the $\psi_0$ waveform
calculated directly from the Teukolsky equation with the $\psi_0$ waveform
obtained by using the Starobinsky-Teukolsky identity on $\psi_4$. We also
compute the out-going echo waveform near infinity, using the near-horizon
$\psi_0$ computed directly from the Teukolsky equation and the Boltzmann
boundary condition on the ECO surface. We show that this echo is quantitatively
different (stronger) than the echo obtained using previous prescriptions.
(abridged)
