Deep-water waves in the cochlea.

In order to obtain physical insight from a mathematical solution of the cochlear mechanics problem, a delicate balance is required between simplifications and refinements in modeling. For the study of the transition of long to short waves (deep-water waves) a closed-form solution is advantageous; this can, however, only be obtained at the cost of further simplification. In previous work the course of the impedance function z(x) was, therefore, reduced to the extreme: in the neighbourhood of the resonance location z(x) was assumed to be a linear function of x - the so-called 'straight-line approximation'. This restriction is removed in the present paper. A 'hyperbolical approximation' of the impedance function z(x) is introduced and it is shown that with this function the two-dimensional cochlea model can be solved in closed form. The computation results show that, for not too large values of the damping parameter delta, the response in the neighbourhood of the resonance location is almost as well represented by the formerly used 'straight-line approximation' as by the 'hyperbolic approximation'. Hence the principal aspects of cochlear resonance are well brought to light with the 'straight-line approximation'. This implies that in the case under consideration the dominant part played by short waves is confirmed. When a larger range of x values is to be considered, the hyperbolic approximation is advantageous. The computed response functions agree better with experimental data from the literature. However, it is clear that really satisfactory agreement seems not possible with a two-dimensional model.