  We present a new covariant, gauge-invariant formalism describing linear
metric perturbation fields on any spherically symmetric background in general
relativity. The advantage of this formalism relies in the fact that it does not
require a decomposition of the perturbations into spherical tensor harmonics.
Furthermore, it does not assume the background to be vacuum, nor does it
require its staticity. In the particular case of vacuum perturbations, we
derive two master equations describing the propagation of arbitrary linear
gravitational waves on a Schwarzschild black hole. When decomposed into
spherical harmonics, they reduce to covariant generalizations of the well-known
Regge-Wheeler and Zerilli equations. Next, we discuss the general case where
the metric perturbations are coupled to matter fields and derive a new
constrained wave system describing the propagation of three gauge-invariant
scalars from which the complete metric perturbations can be reconstructed. We
apply our formalism to the Einstein-Euler system, dividing the fluid
perturbations into two parts. The first part, which decouples from the metric
perturbations, obeys simple advection equations along the background flow and
describes the propagation of the entropy and the vorticity. The second part
describes a perturbed potential flow, and together with the metric
perturbations it forms a closed wave system.
