  Travelling waves in woodpile chains are typically nanoptera, which are
composed of a central solitary wave and exponentially small oscillations. These
oscillations have been studied using exponential asymptotic methods, which
typically require an explicit form for the leading-order behaviour. For many
nonlinear systems, such as granular woodpile chains, it is not possible to
calculate the leading-order solution explicitly. We show that accurate
asymptotic approximations can be obtained using numerical approximation in
place of the exact leading-order behaviour. We calculate the oscillation
behaviour for Toda woodpile chains, and compare the results to exponential
asymptotics based on tanh-fitting, Pad\'{e} approximants, and the adaptive
Antoulas-Anderson (AAA) method. The AAA method is shown to produce the most
accurate predictions of the amplitude of the oscillations and the mass ratios
for which the oscillations vanish. This method is then applied to study
granular woodpile chains, including chains with Hertzian interactions -- this
method is able to calculate behaviour that could not be accurately approximated
in previous studies.
