Foundations of the WKB Approximation for Models of Cochlear Mechanics in 1- and 2-D

TL;DR

This paper provides a comprehensive tutorial on the mathematical foundations, derivation, and implementation of the WKB approximation for cochlear mechanics models in 1-D and 2-D, highlighting its analytical advantages over numerical methods.

Contribution

It offers the first rigorous exposition and detailed implementation guidance of the WKB approximation in cochlear mechanics modeling.

Findings

WKB approximation simplifies cochlear mechanics analysis with closed-form equations.

It enables exploration of frequency tuning and active amplification phenomena.

The tutorial includes software implementation details for 1-D and 2-D models.

Abstract

The Wentzel-Kramers-Brillouin (WKB) approximation is frequently used to explore the mechanics of the cochlea. As opposed to numerical strategies, the WKB approximation facilitates analysis of model results through interpretable closed-form equations, and can be implemented with relative ease. As a result, it has maintained relevance in the study of cochlear mechanics for half of a century. Over this time, it has been used to study a variety of phenomena including the limits of frequency tuning, active displacement amplification within the organ of Corti, feedforward mechanisms in the cochlea, and otoacoustic emissions. Despite this ubiquity, it is challenging to find rigorous exposition of the WKB approximation's formulation, derivation and implementation in cochlear mechanics literature. In this tutorial, I discuss the foundations of the WKB approximation in application to models of cochlear macromechanics in 1-D and 2-D. This includes mathematical background, rigorous derivation and details of its implementation in software.