H. Y. Alkahby, M. A. Mahrous, & B. Mamo
Abstract:
In this paper we present two mathematical models for the
basilar membrane. In the first model the membrane is represented as an
annular region. In the second model the basilar membrane is treated as
a rectangular region. Comparison of the two models allows us to study
the effect of the curvature of the basilar membrane on the range of
the frequencies of hearing. The differential equation of both models
is a fourth order partial differential equation derived from the
classical plate theory. Boundary conditions are defined as a region
with four sides. The conditions are different on each side and
together form an interesting physiological combination, relative to
standard engineering problems. Eigenvalues of the differential
equations of the two models are obtained numerically. A comparison of
the eigenvalues of the two models clearly shows that the range of the
hearing frequencies of the first model is larger than that of the
second model. The results indicate strongly that the curvature of the
basilar membrane plays an important role in the hearing process.
Curvature and measurement of curvature should be allowed in future
models and experiments of the inner ear.
Published January 21, 2000.
Subject classfications: 92C05, 92610, 35G15, 34B10.
Key words: Basilar membrane, eigenvalue, hearing frequencies.
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M. A. Mahrous
Department of Mathematics, University of New Orleans
New Orleans, LA 70148, USA