  We study momentum and energy propagation in 1D tapered chains of spherical
granules which interact according to a Hertz potential. In this work we apply
the binary collision approximation, which is based on the assumption that
transfer of energy along the chain occurs via a succession of two-particle
collisions. Although the binary theory correctly captures the trends of
increase or decrease of kinetic energy and momentum, the actual values of these
quantities are not in good quantitative agreement with those obtained by
numerically integrating the full equations of motion. To address this
difficulty we have developed a mixed numerical/analytical correction algorithm
to provide an improved estimate of the velocity of the particles during pulse
propagation. With this corrected velocity we are in turn able to correctly
predict the momentum and kinetic energy along the chain for several tapering
configurations, specifically for forward linear, forward exponential, backward
linear and backward exponential tapering.
