  We use exponential asymptotics to study travelling waves in woodpile systems
modelled as singularly perturbed granular chains with zero precompression and
small mass ratio. These systems are strongly nonlinear, and there is no
analytic expression for their leading-order solution. We instead obtain an
approximated leading-order solution using a hybrid numerical-analytic method.
We show that travelling waves in these nonlinear woodpile systems are typically
"nanoptera", or travelling waves with exponentially small but non-decaying
oscillatory tails which appear as a Stokes curve is crossed. We demonstrate
that travelling wave solutions in the zero precompression regime contain two
Stokes curves, and hence two sets of tailing oscillations in the solution. We
calculate the behaviour of these oscillations explicitly, and show that there
exist system configurations which cause the oscillations to cancel entirely,
producing solitary wave behaviour. We then study the behaviour of travelling
waves in woodpile chains as precompression is increased, and show that there
exists a value of the precompression above which the two Stokes curves coalesce
into a single curve, meaning that cancellation of the tailing oscillations no
longer occurs. This is consistent with previous studies, which showed that
cancellation does not occur in chains with strong precompression.
