e Bogoliubov dispersion relation [@BEC-review] $\omega({\boldsymbol{k}}) =
k\sqrt{1+k^2/4}$ (with $c=\xi=1$). Note that the smallest velocity, $c=1$, is that of long waves, and that waves of smaller wavelength travel faster. For $\gamma>0$, the planar waves are dampened out and the dispersion relation becomes $\omega({\boldsymbol{k}})=-i\gamma (k^2/2+1)+\sqrt{k^2+k^4/4-\gamma^2}$. The damping rate is determined by $\gamma$ and increases quadratically with the wavenumber. Also, in this case all waves have a group velocity faster than a minimum one that for small $\gamma$ is close to $c=1$.

When $V_p<1$ all the waves escape the neighborhood of the impurity (see an example in Fig. \[fig:density\_gamma\](a)) and are dissipated in a boundary buffer region that has large $\gamma$ (see numerical details in the Appendix and Supplemental Material [@SupplMat]). After a transient the condensate achieves a steady state when seen in the frame comoving with the impurity. Fig. \[fig:density\_gamma\](b) shows a steady spatial configuration for $\gamma=0$ and $V_p=0.9$. Figs. \[fig:density\_gamma\](d-e) show different profiles of the condensate density along the $x$ direction across the impurity position for $V_p$. The condensate density is depleted near the impurity due to the repulsive interaction, and its general shape depends on the speed $V_p$ and thermal drag $\gamma$. If $\gamma=0$ and $V_p\leq 1$ the density of this steady state has a rear-front symmetry with respect to the particle position (see specially Fig. \[fig:density\_gamma\](d)), so that under integration in Eq. (\[eq:fp2\]) the net force is zero. The presence of dissipation ($\gamma>0$) breaks this symmetry even if $V_p<1$ so that a net drag will appear in agreement with the calculation of Sect. \[sec:drag\]. When $V_p>1$ there are waves that can not escape from the neighborhood of the impurity, forming fringes in front of it and a wake behind it. (see Fig. \[fig:density\_gamma\](c) and \[fig:density\_gamma\](f) and Supplemental material [@SupplMat]). Similar 