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Physics and Astronomy Department
Physics and Astronomy Comps Papers
Carleton College Year
A physicist’s guide to the ear
Andrew J. P. Fink
This paper is posted at Digital Commons@Carleton College.
A physicist’s guide to the ear
Andrew J. P. Fink
April 30, 2004
The ear is able to detect sound energies spanning 12 orders of magnitude at frequencies ranging from 20-20,000 Hz. This paper discusses the physical mechanisms underlying the stunning capabilities of the auditory system. We follow a sound wave from the external ear to transduction into an electro-chemical signal for processing by the brain.
We discuss the middle ear in terms of impedance matching and derive an expression for sound transmission from one medium to another. We focus on the inner ear and pay particular attention to the function of the basilar membrane and the properties of the traveling wave both in terms of differential equations that describe the system as well as the impedance of the membrane itself. Finally, we examine the properties of the outer and inner hair cells with particular focus placed on the motile properties of the outer hair cells. We explain the mechano-electrical transduction of the inner hair cells.
Motion appears in many aspects—but there are two obvious kinds, one which appears in astronomy and another which is the echo of that. As the eyes are made for astronomy so are the ears made for the motion which produces harmony.
Plato From the first observations of the cochlear partition it was clear that they represented a system about which physical science provided little knowledge and that many years would be required to understand clearly.
Georg von Békésy You can observe a lot by watching.
Yogi Berra Introduction Organisms that can detect sound posses a huge advantage over those that cannot.
The detection of sound, however, is very complicated, requiring an exquisitely designed detector. Evolution commands amazing power and when life literally depends on the quality of an organism’s sound detector, tremendous amounts of time and energy can be devoted to the problem of making one. Thus, Nature has developed exquisitely sensitive biological detectors, whose incredible precision is matched by daunting complexity.
Indeed, the auditory system boasts a tantalizing collection of problems for the physicist:
the ear must convert a sound wave into an electro-chemical signal. It must first transmit that sound from an air medium to a fluid medium and then filter it and amplify it to prepare it for mechano-electric transduction. Finally, it must transduce the mechanical energy into the electro-chemical language of the nervous system. This whole process must occur more or less instantaneously and in fact, the ear is one of the fastest responding sensory mechanisms. Furthermore, the sound must be faithfully recorded, but also compressed to fit the information constraints applied by the nervous system: there are only so many neurons and they can only fire so fast. Finally, detection must occur over frequencies from 20Hz to 20kHz and over intensities spanning 12 orders of magnitude!
In this paper we will follow a sound wave from the environment’s air medium to its electro-mechanical transduction into a nerve signal, focusing specifically on the transmission of sound across the air-fluid interface and the subsequent filtering and amplification of that sound that occurs in the inner ear. Finally, we will briefly discuss mechano-electric transduction.
Physiology and function of the outer and middle ear Getting sound into the body In this paper we will think of sound as periodic fluctuations in the pressure of an elastic medium, or more simply, the periodic movement of the molecules that make up an elastic medium. We therefore must first determine how the ear transfers periodic movement of air molecules in the environment to periodic movements of the fluid inside the body, the job of the outer and middle ears (Fig. 1).
Figure 1: A diagram of the hearing system consisting of the outer, middle, and inner ears. We will focus on the eardrum: the bridge between the outer and middle ears; the middle ear bones (here labeled Ossicular chain): impedance matchers that transmit sound from the outer to inner ears; and the cochlea: a fluid filled tube (see inset) responsible for transduction of the sound signal into an electro-chemical signal.1 The outer and middle ears are mechanical vibrators and amplifiers; the former gathers sound and the latter transmits and transforms it. When a sound wave enters the ear, it propagates to the eardrum, imparting kinetic energy to it in the form of mechanical vibrations. Connected to the eardrum is a chain of three bones that bridge the space from the outer ear to the inner ear. It serves as the solution to the problem of getting sound waves that were once in air into a fluid-filled body.
These bones serve to transmit sound from the air medium of the environment to the fluid medium of the inner ear. As we will show in the following section this task is not trivial and represents the ear’s first major challenge.
Why is it so hard to transmit sound from an air to a fluid medium?
We want to find out how the properties of two adjacent media affect the power transferred from one medium to the next. Let’s do this by thinking about the pressure waves in the two media. Assume there is some incident pressure wave pi, a reflected
pressure wavep r, and a transmitted pressure wave pt defined as follows:
Now assume that we know Pi and want to find Pr and Pt. Let’s also assume that the
following boundary conditions exist across the boundary between the two media:
1. The power incident on the boundary is equal to the power reflected off of the boundary plus the power transmitted through the boundary.
2. The velocity across the boundary is continuous.
Since pi, pr, and pt completely describe the pressure wave and therefore the pressure in the system, because of our first boundary condition we can assume that
giving us one equation for two unknowns and leaving us to find a second equation relating our pressure-wave amplitudes.
To do this we must introduce the concept of acoustical impedance, a term that will be critical in our discussion of the ear in general. Qualitatively, we can think of impedance as resistance to movement and in turn acoustic impedance as resistance to the movement required for sound propagation. Quantitatively we will think of it as the ratio of acoustic pressure to volume velocity,
where Z is the acoustic impedance, A is the area propagating the sound, and v is the velocity of that area.2 We can thus express the velocities of the waves in the following
The acoustical impedance of the incident and reflected waves is the same since they are in the same medium, only the transmitted wave experiences different acoustical impedance. Applying our second boundary condition we find that
We are ultimately interested in the ratio of the powers of the incident pressure wave to the intensity of the transmitted pressure wave. For sound waves the power is given
where I is the power of the sound wave. (10) can be understood by thinking about it in terms of units. Calling upon our definition of Z we can rewrite this
The units of pressure are N/m2, velocity is m/s, and area m2. Multiplying these together yields N m/s, or Watts, the units of power.
Using (10) we can define a transmission coefficient for the ratio of transmitted power as
A plot of this function is shown below (Fig. 2). It is clear that the maximum sound intensity is transferred when the acoustical impedances of the two media are equal.
Figure 2: A plot of our transmission function. Note that where R = 1 the maximum power is transferred.
To either side of this peak the power transfer drops off steeply.
The ratio of the acoustic impedances of water and air is 3880:1.3 Therefore, for the ear R=0.00027 and T=0.001 meaning that 99.9% of the sound would be lost if the ear was a simple air/water interface.
The ear’s solution As we see in Figure 3 the ear is not a simple air/water interface. Indeed, the bone chain of the air-filled middle ear serves to “match” the impedances of the air of the outer ear on one side with the fluid of the inner ear on the other.
Figure 3: The middle ear is an impedance “matcher”, amplifying the incoming sound waves so that they may enter into the fluid-filled inner ear without significant loss of acoustic power.4 The eardrum, the membrane separating the outer and middle ears, is about twenty times larger than the oval window, the membrane separating the middle and inner ears.5 This difference in area is the pressure-amplifying mechanism of the middle ear that allows it to reduce the affects of the impedance mismatch. The bones of the chain act as levers to further amplify the pressure across the middle ear. Thus, the ear overcomes the differences in impedances of air and water and transmits sound into the fluid-filled inner ear.
The inner ear The cochlea The stapes, the final bone in the chain of the middle ear is fused to the oval
vibrations of the stapes cause the oval window to vibrate, resulting in fluid displacement in the cochlea. A cross section of the cochlea (Fig. 4) shows that it is divided into three, fluid-filled chambers, which for our purposes we will think of as two chambers, reducing the upper two to a single undivided chamber. The basilar membrane separates these two chambers. We will study its oscillations in detail. Atop the basilar membrane sits the organ of Corti, responsible for sound transduction and detection. The organ of Corti runs the entire length of the cochlea, always resting on the top of the basilar membrane.
Figure 4: The final bone in the bone chain of the middle ear, the stapes, is fused to the oval window, which opens into the cochlea. Stapes movement results in fluid flow within the cochlea (see arrows in figure) that ultimately results in sound detection. A cross section of the cochlea is shown in the lower right corner. An important structure to note is the basilar membrane, the strip that divides the lower chamber from the upper section of the cochlea. Atop the basilar membrane sits the organ of Corti.6 The basilar membrane If we stretch out the cochlea we can think of the upper chamber and the lower chamber as long sections separated by the thin basilar membrane (Fig. 5). The stapes is attached to the upper section at the oval window. The lower section ends at the round window, below the stapes. We need to understand how this strip, the basilar membrane, moves if we want to begin to understand the inner ear’s methods of sound transduction.
Figure 5: Looking into a cross section of the cochlea. The arrows here indicate how the fluid flow caused by the movement of the stapes affects the basilar membrane, the strip running up through the center of the cochlea. We will think of the basilar membrane as a simple strip separating two fluid filled chambers and we will discuss how fluid movement determines the movement of the membrane.7 The basilar membrane’s movement directly affects the movement of the organ of Corti, the ear’s mechano-electric transducer. Thus, understanding basilar membrane movement is critical to understanding the function of the hearing system. We will thus devote the next main section of the paper to an analysis of basilar membrane function.
Basilar membrane function This section constitutes the main thrust of the paper. I am going to discuss the development of our theories of basilar membrane function, leaving out most of the details and less significant theories and focusing primarily on the key ideas that provide a basis for our current understanding of the ear. By the end we will have a model of how the basilar membrane responds to fluid movement in the cochlea.
Helmholtz and the resonance place theory During the winter of 1857 Herrmann von Helmholtz gave a lecture at Bonn in which he presented a new theory of hearing based on resonance and Ohm’s acoustic law, the relatively new ideas about acoustic waves published by Ohm and based on the work of Fourier.8 Helmholtz thought of the basilar membrane as a strip of tuned resonators, transversely stretched across the cochlea, sort of like a piano. Each of these resonators had a natural frequency at which it vibrated, and thus when the ear was stimulated with a signal of, 440 Hz say, the 440 Hz resonator would vibrate, telling the brain that it had just heard an A. Since Fourier and Ohm had shown that complex waves are nothing more than combinations of pure tones, this collection of resonators would perform a Fourier transform of the waveform, breaking it down into its component parts. Thus Helmholtz thought of a system that could detect essentially any acoustical wave, where the place of stimulation of the basilar membrane determined the frequency component of the wave, and the degree of displacement determined the wave’s amplitude.
There were, however, significant problems with Helmholtz’s theory and there is a clear argument against resonance being the primary means of sound detection. In order to have a resonator as frequency-selective as Helmholtz’s rods are, the resonator must be very lightly damped. The more a resonator is damped, the lower its degree of frequency selectivity. The rods Helmholtz proposed would have to have had practically no damping if they were to posses the kind of frequency selectivity in existence in the ear. However, the less a resonator is damped, the more it resonates. Thus, if we were to have lightly damped resonators in our basilar membrane, they would ring far longer than is observed.