de Boer E, Viergever M A
Hear Res. 1984 Feb;13(2):101-12. doi: 10.1016/0378-5955(84)90101-1.
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.
通过半解析和渐近解法可以深入了解耳蜗力学，其中刘维尔 - 格林（LG）方法——在另一种情况下也被称为WKB方法——是最重要的一种。本文从一般角度描述了流体波的色散特性，并在此基础上推导了LG公式。结果表明，LG方法的程函方程与色散波理论中的色散关系相同。对群速度的考虑进而导出了核心LG公式，该公式曾在一篇关于LG方法的早期论文中使用过（Boer, E.和Viergever, M.A.（1982）：《听觉研究》8, 131 - 155）。该公式似乎同样适用于耗散系统和有源（即能量产生）系统。在众多可能的相关主题中，选取了两个进行更深入的讨论：能量的放大、集中和扩展，以及耳蜗波的反射问题。在后一个问题中，阐述了为什么耳蜗波尽管表现出很大程度的色散却不发生反射，以及在何种条件下不发生反射。该分析揭示了模型在波传播方向上的一个基本不对称性：与正常方向相反传播的波可能会发生反射，而沿正常方向传播的波则不会反射。
