Many details of the operation of the mammalian cochlea are known, but how they all work together to produce cochlear amplification is not understood. Outer-hair-cell (OHC) motility produces two kinds of amplification: non-propagating amplification (NPA) that is from local OHCs, and traveling-wave amplification (TWA) that increases basilar-membrane (BM) motion. Proposed here are a series of hypotheses that provide a new explanation, the “OoC-area-pump”, for TWA: (1) In the short-wave region OHC vibrations cause cyclic longitudinal motion of fluid in the organ of Corti (OoC) and peri‑Deiters-cell tissue, (2) the longitudinal motion changes the local OoC area, which (3) by reticular-lamina (RL) movement drives the fluid in scala media in a way that amplifies the fluid-pressure traveling wave. (4) At the NPA-TWA changeover frequency, an abrupt change in the OoC frequency-wavenumber relationship is due to positive feedback between TWA and the mode of cochlear motion that is dominant, aided by focusing of the pressure traveling wave. It is hypothesized that OoC radial expansion and radial force from the Deiters-cell phalangeal process act to advance RL and/or lateral-compartment phase. Finally, it is hypothesized that human and lab-animal frequency tuning have similar bandwidths in distance along the cochlea because their traveling-wave wavelengths are similar in the corresponding short-wave regions. Experiments are needed to test these hypotheses and to determine for TWA whether the OoC-area-pump hypothesis replaces or supplements the “OHCs-act-on-BM” hypothesis. Several tests are outlined that can be done with current methodology. A key step in the evolution of mammalian hearing was the development of the complex OoC anatomy, including Deiters cells and OoC fluid spaces that allow local wide-band NPA to produce TWA that enables small local increments of gain to accumulate in the traveling wave and sharpen tuning.