Multiple Scattering of Seismic Waves from Ensembles of Upwardly Lossy Thin Flux Tubes

Our previous semi-analytic treatment of f$f$- and p$p$-mode multiple scattering from ensembles of thin flux tubes (Hanson and Cally, Astrophys. J.781, 125, 2014a; 791, 129, 2014b) is extended by allowing both sausage and kink waves to freely escape at the top of the model using a radiative boundary condition there. As expected, this additional avenue of escape, supplementing downward loss into the deep solar interior, results in substantially greater absorption of incident f$f$- and p$p$-modes. However, less intuitively, it also yields mildly to substantially smaller phase shifts in waves emerging from the ensemble. This may have implications for the interpretation of seismic data for solar plage regions, and in particular their small measured phase shifts.