Ranke revisited—a simple short-wave cochlear model
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An integral equation for the pressure difference across the cochlear partition is derived from the classical assumptions. This equation can be solved to any desired precision by the numerical method of Lesser and Berkley. Alternately, making the usual long-wave assumption, it can be reduced to the second-order differential equation studied by Zwislocki and many others. Although many aspects of this solution agree with observations, the long-wave assumption is of doubtful validity for many interesting frequencies and locations, and the behavior is more sensitive to the size of the scalae than experimental results suggest. A third approach is to make the short-wave assumption proposed by Otto Ranke. The integral equation then can be transformed into a pair of first-order differential equations which can easily be solved in closed form. The resulting simple formulas can be fit to most of the observations of Békésy and others. In addition, the short-wave theory leads to simple explanations of “paradoxical motion” and the effects of bone conduction. The phase characteristics of the theory, however, show major deviations from the experiment—a failure which, it is believed, is due not to the short-wave approximation per se, but rather to the basic physical simplifications common to virtually all cochlear theories.In this article, we present new results on simple points, minimal non-simple sets (MNS) and P-simple points. In particular, we propose new characterizations which hold in dimensions 2, 3 and 4, and which lead to efficient algorithms for detecting such points or sets. This work is settled in the framework of cubical complexes, and some of the main results are based on the properties of critical kernels.
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