Aims. The aim of this work is to demonstrate that the probe-to-spacecraft potential measured by RPW on Solar Orbiter can be used to derive the plasma (electron) density measurement, which exhibits both a high temporal resolution and a high level of accuracy. To investigate the physical nature of the solar wind turbulence and waves, we analyze the density and magnetic field fluctuations around the proton cyclotron frequency observed by Solar Orbiter during the first perihelion encounter (similar to 0.5AU away from the Sun). Methods. We used the plasma density based on measurements of the probe-to-spacecraft potential in combination with magnetic field measurements by MAG to study the fields and density fluctuations in the solar wind. In particular, we used 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. Results. We find that the density fluctuations are 180 degrees out of phase (anticorrelated) with the compressible component of magnetic fluctuations for intervals of turbulence, whereas they are in phase for the circular-polarized waves. We analyze, in detail, two specific events with a simultaneous presence of left- and right-handed waves at di fferent frequencies. We compare the 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 > 10(-6) m(-1), which corresponds to a wavelength of 7 x 10(6) > lambda > 10(6) m. We conclude that it is most likely that both the leftand right-handed waves correspond to the low-wavenumber part (close to the cut-o ff at Omega(cHe++)) of the proton-band electromagnetic ion cyclotron (left-handed wave in the plasma frame confined to the frequency range Omega(cHe++) < omega < Omega(cp)) 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.