  We study ``nanoptera'', which are non-localized solitary waves with
exponentially small but non-decaying oscillations, in two singularly-perturbed
Hertzian chains with precompression. These two systems are woodpile chains
(which we model as systems of Hertzian particles and springs) and diatomic
Hertzian chains with alternating masses. We demonstrate that nanoptera arise
from Stokes phenomena and appear as special curves, called Stokes curves, are
crossed in the complex plane. We use techniques from exponential asymptotics to
obtain approximations of the oscillation amplitudes. Our analysis demonstrates
that traveling waves in a singularly perturbed woodpile chain have a single
Stokes curve, across which oscillations appear. Comparing these asymptotic
predictions with numerical simulations reveals that this accurately describes
the non-decaying oscillatory behavior in a woodpile chain. We perform a similar
analysis of a diatomic Hertzian chain, that the nanpteron solution has two
distinct exponentially small oscillatory contributions. We demonstrate that
there exists a set of mass ratios for which these two contributions cancel to
produce localized solitary waves. This result builds on prior experimental and
numerical observations that there exist mass ratios that support localized
solitary waves in diatomic Hertzian chains without precompression. Comparing
asymptotic and numerical results in a diatomic Hertzian chain with
precompression reveals that our exponential asymptotic approach accurately
predicts the oscillation amplitude for a wide range of system parameters, but
it fails to identify several values of the mass ratio that correspond to
localized solitary-wave solutions.
