  We derive a family of ideal (nondissipative) 3D sound-proof fluid models that
includes both the Lipps-Hemler anelastic approximation (AA) and the Durran
pseudo-incompressible approximation (PIA). This family of models arises in the
Euler-Poincar\'{e} framework involving a constrained Hamilton's principle
expressed in the Eulerian fluid description. The derivation in this framework
establishes the following properties of each member of the entire family: the
Kelvin-Noether circulation theorem, conservation of potential vorticity on
fluid parcels, a Lie-Poisson Hamiltonian formulation possessing conserved
Casimirs, a conserved domain integrated energy and an associated variational
principle satisfied by the equilibrium solutions. \smallskip
  Having set the stage with the derivations of 3D models using the constrained
Hamilton's principle, we then derive the corresponding 2D vertical slice models
for these sound-proof theories.
