An integrable model for the basilar membrane

Assuming a logarithmic relationship between place and characteristic frequency, proportionality between the latter and phase velocity, and a loss factor independent of place, the wave equation for the basilar membrane displacement can be integrated analytically. The main theoretical results are as follows: (1) The low-frequency phase slope equals approximately −Kωr(x), where K is a dimensionless constant and ωr(x) is the characteristic frequency at place x; (2) the asymptotic high-frequency amplitude slope equals −6(K+1) dB/oct; (3) just above resonance the phase reaches a minimum value of about −K rad. The Rhode (1971) low-frequency phase-slope data yields K=27±3. The asymptotic high-frequency amplitude slope should therefore be (168±18) dB/oct. Surprisingly, the great frequency selectivity of the cochlea as reflected in the large numerical value of the high-frequency amplitude-slope is not determined by hydromechanical losses, which primarily influence behavior, near resonance, but the constant K whose physical meaning is approximately “2π times the number of wavelengths” on the membrane for low frequencies. With the above value for K, the number of wavelengths becomes about five—considerably more than found by von Békésy. Nevertheless, it is interesting to note that the phase slopes and high-frequency amplitude slopes of von Békésy (1960, pp. 446–463) are not inconsistent the light of the present theoretical calculations. If the membrane becomes stiffer at the high amplitudes at which von Békésy made his measurement, both phase and amplitude slopes would be decreased.