Heterogeneity can be accounted for by a random potential in the wave equation. For acoustic waves in a fluid with fluctuations of both density and compressibility (as well as for electromagnetic waves in a medium with fluctuation of both permittivity and permeability) the random potential entails a scalar and an operator contribution. For simplicity, the latter is usually overlooked in multiple scattering theory: whatever the type of waves, this simplification amounts to considering the Helmholtz equation with a sound speed $c$ depending on position $mathbf{r}$. In this work, a radiative transfer equation is derived from the wave equation, in order to study energy transport through a multiple scattering medium. In particular, the influence of the operator term on various transport parameters is studied, based on the diagrammatic approach of multiple scattering. Analytical results are obtained for fundamental quantities of transport theory such as the transport mean-free path $ell^*$, scattering phase function $f$ and anisotropy factor $g$. Discarding the operator term in the wave equation is shown to have a significant impact on $f$ and $g$, yet limited to the low-frequency regime i.e., when the correlation length of the disorder $ell_c$ is smaller than or comparable to the wavelength $lambda$. More surprisingly, discarding the operator part has a significant impact on the transport mean-free path $ell^*$ whatever the frequency regime. When the scalar and operator terms have identical amplitudes, the discrepancy on the transport mean-free path is around $300,%$ in the low-frequency regime, and still above $30,%$ for $ell_c/lambda=10^3$ no matter how weak fluctuations of the disorder are. Analytical results are supported by numerical simulations of the wave equation and Monte Carlo simulations.
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