  We extend the general relativistic Lagrangian perturbation theory, recently
developed for the formation of cosmic structures in a dust continuum, to the
case of model universes containing a single fluid with a single-valued analytic
equation of state. Using a coframe-based perturbation approach, we investigate
evolution equations for structure formation in pressure-supported irrotational
fluids that generate their rest-frame spacetime foliation. We provide master
equations to first order for the evolution of the trace and traceless parts of
barotropic perturbations that evolve in the perturbed space, where the latter
describes the propagation of gravitational waves in the fluid. We illustrate
the trace evolution for a linear equation of state and for a model equation of
state describing isotropic velocity dispersion, and we discuss differences to
the dust matter model, to the Newtonian case, and to standard perturbation
approaches.
