We use the plasma density based on measurements of the probe-to-spacecraft potential in combination with magnetic field measurements by MAG to study fields and density fluctuations in the solar wind observed by Solar Orbiter during the first perihelion encounter ($sim$0.5~AU away from the Sun). In particular we use the polarization of the wave magnetic field, the phase between the compressible magnetic field and density fluctuations and the compressibility ratio (the ratio of the normalized density fluctuations to the normalized compressible fluctuations of B) to characterize the observed waves and turbulence. We find that the density fluctuations are out-of-phase with the compressible component of magnetic fluctuations for intervals of turbulence, while they are in phase for the circular-polarized waves around the proton cyclotron frequency. We analyze in detail two specific events with simultaneous presence of left- and right-handed waves at different frequencies. We compare observed wave properties to a prediction of the three-fluid (electrons, protons and alphas) model. We find a limit on the observed wavenumbers, $10^{-6} < k < 7 times 10^{-6}$~m$^{-1}$, which corresponds to wavelength $7 times 10^6 >lambda > 10^6$~m. We conclude that most likely both the left- and right-handed waves correspond to the low-wavenumber part (close to the cut-off at $Omega_{cmathrm{He}++}$) proton-band electromagnetic ion cyclotron (left-handed wave in the plasma frame confined to the frequency range $Omega_{cmathrm{He}++} < omega < Omega_{cmathrm{H}+}$) waves propagating in the outwards and inwards directions respectively. The fact that both wave polarizations are observed at the same time and the identified wave mode has a low group velocity suggests that the double-banded events occur in the source regions of the waves.
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