RELAXATION THEORY OF HEARING

In the currently generally accepted theory of hearing, the organ of Corti and the basilar membrane, located in the inner ear, are regarded as a system of elastic resonators—something like a piano, whose strings are vibrated by the transverse fibers that excite, in their oscillations, the endings of the auditory nerve.

In a recent work Ya. I. Frenkel points out the untenability of this theory from a purely physical point of view. He believes that elastic const—

Relaxation Theory of Hearing

the constants of nervous and muscular tissues are too small to provide natural frequencies of oscillation of the fibers comparable with sound frequencies. This does not permit one to speak of ordinary resonance. Instead the author proposes a new mechanism of sound perception, based on the “quasi-resonant” character of the absorption of energy by a system experiencing a frictional force proportional to velocity (such a frictional force arises, for example, when the system is immersed in a viscous liquid, in the medium that fills the inner ear). As is known, such a system may be characterized by the so-called relaxation time \(\tau\) (that is, the time required for the system to return to its normal state after the removal of external forces). The time dependence of the effect produced in such a system by an external force of the form \(\operatorname{Re}\{F_0 e^{i\omega t}\}\) is given by the expression

\[ \operatorname{Re}\left\{AF_0 \frac{e^{i\omega t}}{1+i\omega\tau}\right\}. \tag{1} \]

(\(F_0\) and \(A\) are real constants.) Accordingly, the work of the force per unit time is proportional to

\[ (AF_0)^2 \frac{\omega\tau}{1+\omega^2\tau^2}. \]

This expression has a maximum at \(\omega=\frac{1}{\tau}\), i.e., the absorption of energy is of a resonant character. It should be noted that the elastic properties of the system are here completely neglected.

The author believes that biological tissue is precisely such a system. This is confirmed, in his opinion, by the properties of other organic substances. For example, for rubber \(\tau \approx 10^{-7}\ \text{sec}\), while the natural frequency of oscillations is only about \(1/50\ \text{sec}^{-1}\). Thus it is assumed that the transverse fibers of the basilar membrane (the author writes instead of them about the endings of the auditory nerve—this is an obvious slip) differ not in their natural frequencies of oscillation (which are all very small and may be equated to zero), but in the values of the relaxation times (the latter vary within the limits, approximately, from \(1/80\ \text{sec}\) to \(10^{-4}\ \text{sec}\)). It is possible, the author notes, that this is connected with the different lengths of the nerve fibers. Owing to the above-mentioned “quasi-resonant” properties of such a system, sound waves of a given frequency are absorbed mainly by only one small group of fibers. The author further remarks that, unlike resonance theory, the new mechanism gives rather broad absorption maxima. In this case the decrease of the effect with deviation of the frequency from \(1/\tau\) is determined by the ratio

\[ \frac{|\omega-1/\tau|}{1/\tau}, \]

and not by the difference \(|\omega-\omega_0|\) (\(\omega_0\) is the natural frequency of the fiber), as before. Consequently, a definite tonal interval corresponds to a constant ratio

\[ \frac{\Delta\omega}{\omega}=\Delta \ln \omega, \]

and thus, with an exponential change in the frequency of the exciting oscillations, the perceived tone changes uniformly. This is a well-known experimental fact (see 2).

The new theory of Ya. I. Frenkel undoubtedly deserves attention and further development.

Cited Literature

1. Ya. I. Frenkel, DAN, new series, No. 4 (1943).
2. S. N. Rzhevkin, Hearing and Speech in the Light of Modern Research, 2nd ed., ONTI, 1936.

V. Averbakh