  We study the hydrodynamic interaction between two closely-spaced waving
elastic cylinders immersed within a viscous liquid, at the creeping flow
regime. The cylinders are actuated by a forced oscillation of the slope at
their clamped end and are free at the opposite end. We obtain an expression for
the interaction force and apply an asymptotic expansion based on a small
parameter representing the ratio between the elastic deflections and the
distance between the cylinders. The leading-order solution is an asymmetric
oscillation pattern at the two frequencies ($\omega_1,\omega_2$) in which the
cylinders are actuated. Higher orders oscillate at frequencies which are
combinations of the actuation frequencies, where the first-order includes the
$2\omega_1,2\omega_2,\omega_1+\omega_2$, and $\omega_1-\omega_2$ harmonics. For
in-phase actuation with $\omega_1= \omega_2$, the deflection dynamics are
identical to an isolated cylinder with a modified Sperm number. For
configurations with $\omega_1\approx \omega_2$, the $\omega_1-\omega_2$ mode
represents the dominant first-order interaction effect due to significantly
smaller effective Sperm number. Experiments are conducted to verify and
illustrate the theoretical predictions.
