de Boer E
Hear Res. 1980 Aug;3(2):97-108. doi: 10.1016/0378-5955(80)90039-8.
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.
为了从耳蜗力学问题的数学解中获得物理见解，在建模的简化和细化之间需要一种微妙的平衡。对于长波到短波（深水波）过渡的研究，封闭形式的解是有利的；然而，这只能以进一步简化为代价来获得。因此，在先前的工作中，阻抗函数z(x)的过程被简化到了极致：在共振位置附近，z(x)被假定为x的线性函数——即所谓的“直线近似”。本文消除了这一限制。引入了阻抗函数z(x)的“双曲线近似”，并表明利用该函数可以以封闭形式求解二维耳蜗模型。计算结果表明，对于不太 大的阻尼参数δ值，共振位置附近的响应几乎可以由先前使用的“直线近似”和“双曲线近似”同样好地表示。因此，耳蜗共振的主要方面通过“直线近似”得到了很好的揭示。这意味着在所考虑的情况下，短波所起的主导作用得到了证实。当要考虑更大范围的x值时，双曲线近似是有利的。计算得到的响应函数与文献中的实验数据更吻合。然而，很明显，对于二维模型来说，似乎不可能得到真正令人满意的吻合。
