Growth rate spectrum of the hosing instability in the long-beam overdense regime

The hosing instability is still one of the main feasibility risks for plasma wakefield accelerator (PWFA) concepts. Though potential mitigation methods have been discussed extensively in the blow-out regime, less attention has been devoted to hosing in the long-beam, overdense regime\footnote{C. B. Schroeder {\it et al.}, Phys. Rev. E {\bf 86}, 026402 (2012)}$^,$\footnote{J. Vieira {\it et al.}, Phys. Rev. Lett. {\bf 112}, 205001 (2014)}$^,$\footnote{R. Lehe {\it et al.}, Phys. Rev. Lett. {\bf 119}, 244801 (2017)}, which is relevant for PWFA concepts using a long drive bunch and geared towards high-energy physics applications.\\ This work presents a fuller picture of the physics of beam hosing in the overdense regime by focusing on the development of the instability when the initial centroid oscillates at a wavelength different than the plasma wavelength $\lambda_p$. We use theory and particle-in-cell simulations with OSIRIS to show that the growth rate for beam hosing is a function of the centroid perturbation wavelength with a maximum at $\lambda_p$ (similarly to laser hosing\footnote{B. J. Duda {\it et al.}, Phys. Rev. Lett. {\bf 83}, 1978 (1999)}), and that this property can be exploited to conditionally achieve damping instead of amplification of the centroid oscillation.