E3 Middle Ear: Impedance Transmission and Transformation

Topic

The sound pressure reaching the tympanic membrane—already shaped by the auricle and the external auditory canal—causes the membrane to vibrate with a total force determined by the amount of sound pressure received and the surface area of ​​the membrane vibrating in response. This same total force, gathered across the large surface area of ​​the tympanic membrane, is then transmitted to the much smaller surface area of ​​the oval window; there, the stapes—the last of the three ossicles in the chain—pushes directly against the cochlear perilymph. Concentrating the same total force onto a much smaller surface area increases the pressure exerted by that force, just as pressing a finger against a small area exerts more pressure than applying the same force over a large area.

In addition to the difference in surface area between the tympanic membrane and the oval window, the ossicular chain itself provides a second pressure boost through its lever action: the malleus (the first ossicle in the chain) has a longer lever arm than the incus (the second ossicle), with which it articulates. When a longer lever arm pushes a shorter one through this joint, the shorter arm moves with less amplitude but greater force than the longer arm; consequently, the ossicular chain delivers a greater force to the stapes than the force the malleus originally received from the tympanic membrane.

The pressure gain resulting from the difference in surface area between the tympanic membrane and the oval window and the pressure gain resulting from the lever action of the ossicular chain do not simply add together; rather, they multiply, because each acts upon the result produced by the previous one: the ossicular chain applies its own leverage-based gain to the force that has already been concentrated as it traveled from the tympanic membrane to the malleus. Combined, these two gains result in the middle ear delivering pressure to the cochlear perilymph that is approximately thirty decibels higher than the original acoustic pressure reaching the tympanic membrane—a gain that enables the middle ear to match the much lower impedance of air with the much higher impedance of the cochlear perilymph, preventing most of the acoustic pressure from being lost to reflection at the interface between the two media.

The approximate thirty-decibel gain achieved by the middle ear is not uniform across all frequencies: the relationship between the acoustic pressure entering the tympanic membrane and the transmitted pressure ultimately received by the cochlear perilymph varies according to sound frequency; this frequency-dependent relationship is known as the middle ear transfer function. The middle ear transfer function depends on the mass and stiffness of the tympanic membrane and the ossicular chain itself—structures that respond more readily to certain frequencies than to others, much like the external auditory canal, which, due to its shape, amplifies some frequencies more than others.

By incorporating the gain resulting from the surface area difference between the tympanic membrane and the oval window, the gain provided by the lever action of the ossicular chain, and the frequency-dependent middle ear transfer function, this model allows for the calculation of the pressure ultimately transmitted to the cochlear perilymph—based on the acoustic pressure that originally reached the tympanic membrane—for any given sound frequency. Thus, the mechanical lever model of the tympanic membrane and ossicular chain introduced at the beginning of this description evolves from a mere qualitative explanation of how the middle ear matches air impedance to cochlear perilymph impedance into a tool capable of simulating the actual amount of transmitted pressure received by the cochlear perilymph at each frequency.

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