  The inner ear achieves a wide dynamic range of responsiveness by mechanically
amplifying weak sounds. The enormous mechanical gain reported for the mammalian
cochlea, which exceeds a factor of 4,000, poses a challenge for theory. Here we
show how such a large gain can result from an interaction between amplification
by low-gain hair bundles and a pressure wave: hair bundles can amplify both
their displacement per locally applied pressure and the pressure wave itself. A
recently proposed ratchet mechanism, in which hair-bundle forces do not feed
back on the pressure wave, delineates the two effects. Our analytical
calculations with a WKB approximation agree with numerical solutions.
