In this paper, we conduct a parametric study on the influence of sponge layer strength on temporal eigenvalue problems arising from the one-dimensional wave equation and the linearized Navier-Stokes equations. Sponge layers have shown to stabilize eigenmodes and introduce additional spatial growth to eigenfunctions. As the strength of sponge layers increases, temporal eigenvalues are displaced and the spatial growth rates of their associat- ed eigenfunctions are modified. In both wave and linearized Navier-Stokes equations, the linear relationship between temporal damping and spatial growth can be specified as an approximate dispersion relation. It can also be shown that an over strengthened sponge layer can reect spatially propagating waves. This reection can lead to a destabilization of the otherwise stable eigenspectrum with alteration of eigenfunction wavelengths. We provide an empirical guideline for determining the optimal sponge layer strength and demonstrate the efficacy of our method in the global stability analysis of the linearized Navier-Stokes equations.