  The relativistic generalization of the Newtonian Lagrangian perturbation
theory is investigated. In previous works, the perturbation and solution
schemes that are generated by the spatially projected gravitoelectric part of
the Weyl tensor were given to any order of the perturbations, together with
extensions and applications for accessing the nonperturbative regime. We here
discuss more in detail the general first-order scheme within the Cartan
formalism including and concentrating on the gravitational wave propagation in
matter. We provide master equations for all parts of Lagrangian-linearized
perturbations propagating in the perturbed spacetime, and we outline the
solution procedure that allows one to find general solutions. Particular
emphasis is given to global properties of the Lagrangian perturbation fields by
employing results of Hodge-de Rham theory. We here discuss how the Hodge
decomposition relates to the standard scalar-vector-tensor decomposition.
Finally, we demonstrate that we obtain the known linear perturbation solutions
of the standard relativistic perturbation scheme by performing two steps:
first, by restricting our solutions to perturbations that propagate on a flat
unperturbed background spacetime and, second, by transforming to Eulerian
background coordinates with truncation of nonlinear terms.
