The 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 Wentzel–Kramers–Brillouin-like approximation (Manoussaki et al., 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 existence 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 the detailed geometry of the cochlear duct for a more quantitative result. The present result clearly demonstrates that the spiral geometry and the geometry of the cochlear duct play decisive roles in distributing the wave energy.