Wave propagation and dispersion in the cochlea

Insight into cochlear mechanics can be obtained from semi-analytical and asymptotic solution methods of which the Liouville-Green (LG) method - in another context known as the WKB method - is the most important one. This paper describes the dispersion properties of fluid waves in general terms and develops the LG formulation on that basis. The eikonal equation of the LG method is shown to be identical to the dispersion relation in dispersive-wave theory. Consideration of the group velocity then leads to the derivation of the central LG formula as it has been used in an earlier paper on the LG method (Boer E. and Viergever M.A. (1982): Hearing Res. 8, 131-155). The formulation appears to apply as well to dissipative and active (i.e., energy-producing) systems. Of the many possible collateral subjects two are selected for a deeper discussion: amplification, concentration and expansion of energy, and the problem of reflection of cochlear waves. In the latter context, it is shown why - and under which conditions - cochlear waves are not reflected, despite the large degree of dispersion that they show. The analysis brings to light a fundamental asymmetry of the model regarding the direction of wave travel: waves travelling in the direction opposite to the normal one are likely to undergo reflection, while waves in the normal direction are not reflected.