Correspondence principle in cochlear mechanics.

Abstract

When only long waves play the most important part in the cochlea, the response can be described by a most simplified model, the one-dimensional model. When short waves are to be included, a more complex model is needed. The response then depends on the dimensionality of the model and is much harder to obtain. This applies especially to the region in the neighborhood of the point where the basilar membrane shows resonance. Both two- and three-dimensional models have been studied to assess the effects of short and long waves. The relative importance of the part played by short waves depends on the damping constant (or loss factor )delta associated with the resonance of the basilar membrane (BM). For very small delta a three-dimensional model is really necessary, it cannot be replaced by a model of lower dimensionality. When delta is small, but not too small, the three-dimensional model can be made equivalent to a two-dimensional one, provided the latter is modified ina specific manner. This paper shows why this is so and which conditions have to be met. The two-dimensional model must undergo two modifications to effect this equivalence. The first modification ensures that the model has the same long-wave behavior. In the second place, a specific additional mass ("added mass") reactance should be added to Z(kappa). An expression for the limiting value of delta, above which this correspondence is valid, is given in the paper. A second, larger, limit is presented as well: when delta is above this limit, the responses of both the three-dimensional and the two-dimensional model are equivalent to that of an appropriately chosen one-dimensional model. In this case too, long-wave behavior must be matched and an "added mass" reactance must be included in Z(kappa). This holds true for the entire cochlea including the region of resonance. For both types of transition the amount of "added mass" is given.