  Cochlea is an important auditory organ in the inner ear. In most mammals, it
is coiled as a spiral. Whether this specific shape influences hearing is still
an open problem. By employing a three dimensional fluid model of the cochlea
with an idealized geometry, the influence of the spiral geometry of the cochlea
is examined. We obtain solutions of the model through a conformal
transformation in a long-wave approximation. Our results show that the net
pressure acting on the basilar membrane is not uniform along its spanwise
direction. Also, it is shown that the location of the maximum of the spanwise
pressure difference in the axial direction has a mode dependence. In the
simplest pattern, the present result is consistent with the previous theory
based on the WKB-like approximation [D. Manoussaki, Phys. Rev. Lett. 96,
088701(2006)]. In this mode, the pressure difference in the spanwise direction
is a monotonic function of the distance from the apex and the normal velocity
across the channel width is zero. Thus in the lowest order approximation, we
can neglect the existance of the Reissner's membrane in the upper channel.
However, higher responsive modes show different behavior and, thus, the real
maximum is expected to be located not exactly at the apex, but at a position
determined by the spiral geometry of the cochlea and the width of the cochlear
duct. In these modes, the spanwise normal velocities are not zero. Thus, it
indicates that one should take into account of the detailed geometry of the
cochlear duct for a more quantitative result. The present result clearly
demonstrates that not only the spiral geometry, but also the geometry of the
cochlear duct play decisive roles in distributing the wave energy.
