Sound enters the cochlea as a traveling wave along the basilar membrane. The wave slows as it approaches the resonant point — the location where the membrane's stiffness matches the frequency — and deposits its energy there. This is the basis of frequency selectivity in hearing: each location along the cochlea responds to a specific pitch.
Momi et al. (arXiv:2603.16047) perform an analytic dissection of the cochlear wave equation and discover that the system supports two fundamentally different kinds of eigenmodes, not one.
The first species is familiar: localized resonant modes centered at the characteristic place for each frequency. These modes arise from internal resonance, requiring a mathematical singularity in the wave equation where the forcing frequency matches the local resonant frequency. The energy concentrates at a point. This is the standard cochlear model.
The second species is new: spatially extended modes that span the entire cochlear length. These arise from globally continuous standing-wave solutions — they satisfy the boundary conditions at both ends of the cochlea without requiring any singularity. They do not concentrate energy at a point. They distribute it everywhere.
The two species have different analytic origins. Localized modes require singular point matching — a specific mathematical procedure for connecting solutions across the resonance point. Extended modes emerge from regular continuation — the globally smooth solutions that exist independent of any resonance.
The cochlea is not one instrument. It is two instruments in the same enclosure — a bank of localized resonators that sort frequency by position, and a waveguide that carries information end-to-end without localizing it. Both are present in the same wave equation, arising from different branches of the same mathematical structure. One ear hears locally. The other hears everything at once.