de Boer E, Viergever M A
Hear Res. 1982 Oct;8(2):131-55. doi: 10.1016/0378-5955(82)90071-5.
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.
本文对二维和三维耳蜗模型的刘维尔 - 格林（LG）计算结果与精确解进行了比较。总体而言，二者吻合度相当高。然而，对于某些模型参数的选择，二维和三维LG解在基底膜响应最大振幅位置之外的区域显示出明显误差。这些误差的根源似乎在于二维和三维模型中（复数）LG波数k(x)的非唯一性：求解k(x)所依据的“程函方程”有多个根。为了更深入地研究这个问题，我们考察了x变化时k = k(x)轨迹的性质。当这个根轨迹接近一个复函数Q(k)（它在程函方程中起主要作用）的鞍点之一时，会发现LG解出现不稳定行为。除了这个特定问题外，LG近似非常适合揭示耳蜗中波传播和衰减的机制。分析清楚地表明了在长波和短波情况下，基底膜的响应为何以及如何达到最大值，以及哪些因素导致了响应的转折和快速下降。本文还专门讨论了LG近似与所研究类型的耳蜗模型中波反射缺失之间的关系。
