  A new concept of semi-compressible fluids is introduced for slightly
compressible visco-elastic fluids (typically rather liquids than gasses) where
mass density variations are negligible in some sense, while being directly
controlled by pressure which is very small in comparison with the elastic bulk
modulus. The physically consistent fully Eulerian models with specific
dispersion of pressure-wave speed are devised. This contrasts to the so-called
quasi-incompressible fluids which are described not physically consistently
and, in fact, only approximate ideally incompressible ones in the limit. After
surveying and modifying models for the quasi-incompressible fluids, we
eventually devise some fully convective models complying with energy
conservation and capturing phenomena as pressure-wave propagation with
wave-length (and possibly also pressure) dependent velocity.
