  Alfv\'enic fluctuations, as modelled by the non-linear interactions of
Alfv\'en waves of various scales, are seen to dominate solar wind turbulence.
However, there is also a non-negligible component of non-Alfv\'enic
fluctuations. The Els\"asser formalism, which is central to the study of
Alfv\'enic turbulence due to its ability to differentiate between parallel and
anti-parallel Alfv\'en waves, cannot strictly separate wavemodes in the
presence of compressive magnetoacoustic waves. In this study, we analyse the
deviations generated in the Els\"asser formalism as density fluctuations are
naturally generated through the propagation of a linearly polarised Alfv\'en
wave. The study was performed in the context of a coronal mass ejection (CME)
propagating through the solar wind, which enables the creation of two solar
wind regimes, pristine wind and a shocked CME sheath, where the Els\"asser
formalism can be evaluated. In these two regimes we studied the deviations of
the Els\"asser formalism in separating parallel and anti-parallel components of
Alfv\'enic solar wind perturbations generated by small-amplitude density
fluctuations. We used an ideal 2.5D magnetohydrodynamic (MHD) model with an
adiabatic equation of state. An Alfv\'en pump wave was injected into the quiet
solar wind by perturbing the transverse magnetic field and velocity components.
This wave subsequently generates density fluctuations through the ponderomotive
force. A CME was injected by inserting a flux-rope modelled as a magnetic
island into the quasi-steady solar wind. The presence of density perturbations
creates an approximately 10% deviation in the Els\"asser variables and
reflection coefficient for the Alfv\'en waves as well as a deviation of
approximately 0.1 in the cross helicity in regions containing both parallel and
anti-parallel fluctuations.
