  Galbrun's equation, which is a second order partial differential equation
describing the evolution of a so-called Lagrangian displacement vector field,
can be used to study acoustics in background flows as well as perturbations of
astrophysical flows. Our starting point for deriving Galbrun's equation is
linearized Euler's equations, which is a first order system of partial
differential equations that describe the evolution of the so-called Eulerian
flow perturbations. Given a solution to linearized Euler's equations, we
introduce the Lagrangian displacement as the solution to a linear first order
partial differential equation, driven by the Eulerian perturbation of the fluid
velocity. Our Lagrangian displacement solves Galbrun's equation, provided it is
regular enough and that the so-called "no resonance" assumption holds. In the
case that the background flow is steady and tangential to the domain boundary,
we prove existence, uniqueness, and continuous dependence on data of solutions
to an initial--boundary value problem for linearized Euler's equations. For
such background flows, we demonstrate that the Lagrangian displacement is
well-defined, that the initial datum of the Lagrangian displacement can be
chosen in order to fulfill the "no resonance" assumption, and derive a
classical energy estimate for (sufficiently regular solutions to) Galbrun's
equation. Due to the presence of zeroth order terms of indefinite signs in the
equations, the energy estimate allows solutions that grow exponentially with
time.
