We investigate the damping of Alfv\'en waves generated by the cosmic ray
resonant streaming instability in the context of the cosmic ray escape and
propagation in the proximity of supernova remnants. We consider ion-neutral
damping, turbulent damping and non linear Landau damping in the warm ionized
and warm neutral phases of the interstellar medium. For the ion-neutral
damping, up-to-date damping coefficients are used. We investigate in particular
whether the self-confinement of cosmic rays nearby sources can appreciably
affect the grammage. We show that the ion-neutral damping and the turbulent
damping effectively limit the residence time of cosmic rays in the source
proximity, so that the grammage accumulated near sources is found to be
negligible. Contrary to previous results, this also happens in the most extreme
scenario where ion-neutral damping is less effective, namely in a medium with
only neutral helium and fully ionized hydrogen. Therefore, the standard
picture, in which CR secondaries are produced during the whole time spent by
cosmic rays throughout the Galactic disk, need not to be deeply revisited.Comment: 13 pages, 9 figures, submitted to MNRA