Validity of the Liouville--Green (or WKB) method for cochlear mechanics

This article presents a comparison of Liouville-Green (LG) calculations and exact solutions of 2- and 3-dimensional cochlea models. The agreement is in general quite good. For certain choices of the model parameters, however, the 2- and 3-dimensional LG solutions show appreciable errors in the region just beyond the location of maximum amplitude of the basilar membrane response. The origin of these errors appears to be the non-uniqueness of the (complex) LG wave number k(x) in 2- and 3-dimensional models: the 'eikonal equation' from which k(x) has to be solved has multiple roots. To study this problem somewhat deeper, the properties of the locus of k=k(x) formed when x is varied, are investigated. Erratic behaviour of the LG solution is found to occur when this root locus approaches one of the saddle points of a complex function of k- called Q(k)- which plays the major role in the eikonal equation. Apart from this specific problem, the LG approximation is very well suited to unravel the mechanisms governing wave propagation and attenuation in the cochlea. The analysis shows clearly why and how the response of the basilar membrane builds up to a maximum and which factors cause a turnover and a rapid decrease to occur, in both the long-wave and the short-wave cases. A special discussion is dedicated to the relation between the LG approximation and the absence of wave reflection in cochlea models of the type studied.