The mode-coupling Liouville-Green approximation for a two-dimensional cochlear model.

The Liouville-Green [or Wentzel-Kramers-Brillouin (WKB)] approximation for the two-dimensional cochlear mechanics problem disagrees with the finite-difference solution in the region after the response peak. This disagreement has left doubts about the validity of the Liouville-Green approximation, and has never been satisfactorily explained. In this paper, it is shown that the Liouville-Green approximation fails to satisfy Laplace's equation. A new solution is proposed, called the mode-coupling Liouville-Green approximation, in which energy is coupled into a second wave mode, so as to obey Laplace's equation. The new approximation gives excellent quantitative agreement with the finite-difference solution. Furthermore, it may provide an explanation for a second vibration mode observed in biological cochleas. Also proposed is a high-order formulation of the stapes displacement term, which is necessary to obtain good agreement between the Liouville-Green approximation and finite-difference solutions at low frequencies.