Determination of roots of eikonal equation for WKB solution in cochlear models

Abstract

The WKB-theory-based asymptotic methods have been used to obtain solutions to analytical cochlear models [E. de Boer, Phys. Rep. 105(3), 141–226]. The local wave number for this solution is obtained as the root of the eikonal equation or the dispersion equation. For a three-dimensional rectangular model, multiple complex roots exist, which could give rise to additional waves on the basilar membrane. However, previous results were based on a single root locus determined using initial estimates obtained from the long-wave approximation. In this study, the winding-number integral technique [P. R. Brazier-Smith and J. F. M. Scott, J. Sound Vib. 145(3), 503–510] is used to obtain a definitive identification of the multiple roots and their dependence on the number of acoustic cross modes. A priori information about the characteristics of the dispersion relationship was used to obtain an initial estimate of the search region. Thereafter the region was refined and the roots located using moments of the winding integral and secant iterations. It was found that higher-order roots are strongly evanescent away from the resonance location. Also, as the number of acoustic cross modes of the duct increases, more roots appear. The influence of these roots on the solution will be discussed.