  The cochlea or inner ear has a remarkable ability to amplify sound signals.
This is understood to derive at least in part from some active process that
magnifies vibrations of the basilar membrane (BM) and the cochlear partition in
which it is embedded, to the extent that it overcomes the effect of viscous
damping from the surrounding cochlear fluid. Many authors have associated this
amplification ability to some type of mechanical resonance within the cochlea,
however there is still no consensus regarding the precise cause of
amplification. Our work is inspired by experiments showing that the outer hair
cells within the cochlear partition change their lengths when stimulated, which
can in turn cause periodic distortions of the BM and other structures in the
cochlea. This paper investigates a novel fluid-mechanical resonance mechanism
that derives from hydrodynamic interactions between an oscillating BM and the
surrounding cochlear fluid. We present a model of the cochlea based on the
immersed boundary method, in which a small-amplitude periodic internal forcing
due to outer hair cells can induce parametric resonance. A Floquet stability
analysis of the linearized equations demonstrates the existence of resonant
(unstable) solutions within the range of physical parameters corresponding to
the human auditory system. Numerical simulations of the immersed boundary
equations support the analytical results and clearly demonstrate the existence
of resonant solution modes. These results are then used to illustrate the
influence of parametric resonance on wave propagation along the BM and explicit
comparisons are drawn with results from another two-dimensional cochlea model.
