  The Els\"{a}sser variables are often used in studies of plasma turbulence, in
helping differentiate between MHD waves propagating parallel or anti-parallel
to the main magnetic field. While for pure Alfv\'en waves in a homogeneous
plasma the method is strictly valid, we show that compressible, magnetoacoustic
waves are in general described by both Els\"{a}sser variables. Furthermore, in
a compressible and inhomogeneous plasma, the pure MHD waves (Alfv\'en, fast and
slow) are no longer normal modes, but waves become linearly coupled or display
mixed properties of Alfv\'en and magnetoacoustic nature. These waves are
necessarily described by both Els\"{a}sser variables and therefore the
Els\"{a}sser formalism cannot be used to strictly separate parallel and
anti-parallel propagating waves. Nevertheless, even in an inhomogeneous plasma,
for a highly Alfv\'enic wave the Els\"{a}sser variable corresponding to the
propagation direction appears still dominating. We suggest that for Alfv\'enic
waves, the relative amplitude of Els\"{a}sser variables depends on the local
degree of inhomogeneity and other plasma and wave properties. This finding has
implications for turbulence studies in inhomogeneous and compressible plasmas,
such as the solar corona and solar wind.
