  The propagation of elastic waves in soft materials plays a crucial role in
the spatio-temporal transmission of mechanical signals, e.g. in biological
mechanotransduction or in the failure of marginal solids. At high Reynolds
numbers $Re \gg 1$, inertia dominates and wave propagation can be readily
observed. However, mechanical cues in soft and biological materials often occur
at low $Re$, where waves are overdamped. Not only have low $Re$ waves been
difficult to observe in experiments, their theoretical description remains
incomplete. In this paper, we present direct measurements of low $Re$ waves
propagating in ordered and disordered soft solids, generated by an oscillating
point force induced by an optical trap. We derive an analytical theory for low
$Re$ wave propagation, which is in excellent agreement with the experiments.
Our results present both a new method to characterize wave propagation in soft
solids and a theoretical framework to understand how localized mechanical
signals can provoke a remote and delayed response.
