It has been suggested that the decay constants of tectorial membrane (TM) traveling waves contribute to differences in the tuning of Tectb -/- mutant mice, wild-Type mice, and humans. However, the experiments underlying these results were obtained in vitro, in the absence of cochlear loads. In this work, we analyze effects of cochlear loads on TM traveling waves using a viscoelastic model. Results demonstrate that hair bundle stiffness has little effect on TM waves, the limbal attachment tends to increase the wavelength of TM waves, and viscous loss to subtectorial fluid tends to decrease the wave decay constant. To understand how the TM is able to support traveling waves despite these loads we examined how TM wave decay constants depend on thickness of the TM. We observe that increasing TM thickness tends to moderate all effects of cochlear loads and that as TM thickness increases, the wave decay constant approaches an asymptotic limit that is independent of cochlear loads. For material properties corresponding to both the apical and basal turns of the mouse TM, the asymptotic limit in decay constant is achieved at roughly the physiological thickness-suggesting an evolutionary pressure for the TM to be just thick enough to allow for wave propagation in the presence of subtectorial fluid.