Efficiently solving numerical problems involving wave-like phenomena is an area of active research due to the many areas in which these problems appear, including seismic imaging, electromagnetics, acoustics and elasticity. The linear systems resulting from discretising these problems are generally very large and highly indefinite, so that there is particular need for preconditioners for them; we shall look in particular at Engquist and Ying’s Sweeping Preconditioner. The motivating theory for this preconditioner includes the fact that the Green’s function of the Helmholtz Half-Plane problem admits a low-rank separable expansion on certain thin, separated domains. Previous numerical experiments show that constructing the preconditioner with absorption (i.e. adding a complex part to the wavenumber of the problem) improves the performance of the preconditioner; we discuss new results that show improvements to the motivating theory when absorption is added.