  Recently proposed classical analogs of topological insulators in phononic
lattices have the advantage of much more accessible experimental realization as
compared to conventional materials. Drawn to their potential practical
structural applications, we investigate how disorder, which is generically
non-negligible in macroscopic realization, can attenuate the topologically
protected edge (TPE) modes that constitute robust transmitting channels at zero
disorder. We simulate the transmission of phonon modes in a
quasi-one-dimensional classical lattice waveguide with mass disorder, and show
that the TPE mode transmission remains highly robust ($\Xi\sim1$) in the
presence of uncorrelated disorder, but diminishes when disorder is spatially
correlated. This reduction in transmittance is attributed to the Anderson
localization of states within the mass disorder domains. By contrast, non-TPE
channels exhibit qualitatively different behavior, with spatial correlation in
the mass disorder leading to significant transmittance reduction (enhancement)
at low (high) frequencies. Our results demonstrate how topologically protected
edge modes drastically modify the effect of spatial correlation on mode
localization.
