  In this paper we revisit non-spherical perturbations of the Schwarzschild
black hole in the context of $f(R)$ gravity. Previous studies were able to
demonstrate the stability of the $f(R)$ Schwarzschild black hole against
gravitational perturbations in both the even and odd parity sectors. In
particular, it was seen that the Regge-Wheeler and Zerilli equations in $f(R)$
gravity obey the same equations as their General Relativity counterparts. More
recently, the 1+1+2 semi-tetrad formalism has been used to derive a set of two
wave equations: one for transverse, trace-free (tensor) perturbations and one
for the additional scalar modes that characterise fourth-order theories of
gravitation. The master variable governing tensor perturbations was shown to be
a modified Regge-Wheeler tensor obeying the same equation as in General
Relativity. However, it is well known that there is a non-uniqueness in the
definition of the master variable. In this paper we derive a set of two
perturbation variables and their concomitant wave equations that describe
gravitational perturbations in a covariant and gauge invariant manner. These
variables can be related to the Newman-Penrose (NP) Weyl scalars as well as the
master variables from the 2+2 formalism.
