# Engineering Acoustics/Conductive Mechanisms of Hearing

!Thumb\|right\|border\|400px\|Anatomy of the Human
Ear

In this section we will discuss the pathways for conduction of acoustic
sound waves to the inner ear. Two methods of conduction will be covered;
first we will cover conduction to the inner ear through the outer and
middle ear then conduction to the inner ear through bone conduction.
Without these conductive pathways the acoustic waves that transmit sound
would not be able to reach the inner ear, and we would therefore be
unable to hear.

## Outer Ear

The first area where sound enters the ear is in the outer ear which can
be broken down into two parts, the Pinna (visible portion of the ear)
and the ear canal.

### Pinna

It was once thought that the Pinna funneled sound into the ear canal,
but the primary function of the Pinna is to aid in sound source
localization. The various ridges in the Pinna filter the signal at
frequencies over 4000 Hz according to the direction of the sound.The
resulting spectral variations allow our brain to determine the elevation
of a sound and localize sounds in reference to our position.[^1]

### Ear Canal

The ear canal can be modeled as a tube approximately 2.8 cm long that is
closed at one end by the tympanic membrane. Unlike a rigid tube that has
a sharp spike in amplitude at the resonant frequency, the ear canal has
a wide resonance peak from approximately 2--5 kHz.[^2] It is important
to remember the ear canal is not rigid nor is not a straight tube, and
therefore damping is introduced.While damping is introduced to the
system, thereby altering the resonant frequency, the important thing is
that the ear canal can raise the Sound Pressure
Level (SPL) of the ear by up to 15 dB
which will amplify the incoming acoustic signal.[^3]

!Thumb\|center\|border\|400px

## Middle Ear

The primary purpose of the middle ear is to act as an impedance-matching
transformer that will allow for acoustic energy to be efficiently
transferred from the air filled outer ear to the liquid filled cochlea.
Knowing the impedance ratio, r, of the liquid in the cochlea to the air
is 4000:1 we can use the following equation to determine what the energy
transmission coefficient, T, would be without the middle ear

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```
$T = \frac{4r}{ \left( r+1 \right) ^2}$

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which gives us a transmission of 0.001 or 0.1%. This transmission value
is equivalent to a SPL drop of around 40 dB.[^4]

To overcome this impedance mismatch,the middle ear employs three
mechanical amplification systems; the areal ratio between the Tympanic
Membrane and the
Stapes, the areal ratio of the
stapes to the Oval Window
the lever effect of the Ossicular
Chain.

### Areal Ratio

While the ossicular chain helps to increase the pressure of the incoming
acoustic signal, the majority of gain in the middle ear is due to the
areal ratio of the tympanic membrane and the stapes. The average area of
the tympanic membrane is 66 $mm^2$ while that of the stapes is 3.2
$mm^2$. The areal ratio between the two areas should therefore be
approximately 20:1, but as mentioned earlier the effective area of the
tympanic membrane is only 65% so, 66 $mm^2$ x 65% = 42.9 $mm^2$. This
results in an areal ratio of 13.4.

!Thumb\|center\|border\|375px

### Tympanic membrane

The tympanic membrane plays an important role in the conduction of
sound. Serving as a
Transducer, the membrane
converts acoustic pressure waves into mechanical motion. Experiments
have shown that the tympanic membrane is not uniform, but has a
spatially variable stiffness. Due to these variations, the effective
area of the membrane is only 65%.[^5] It is important to mention that
the stiffness of the tympanic membrane is also very important to the
efficient transfer of energy as being too stiff would reflect a large
amount of energy back through the ear canal, while being too flacid
would cause the membrane to absorb too much energy.[^6]

Another Important contribution of the Tympanic membrane is the gain
contributed by the curved membrane principle. This model represents the
membrane in two sections with the manubrium at the center. Due to the
curved membrane principle, the force on the manubrium is greater than on
the membrane and thus the strength of the incoming signal is increased
by a factor of 2.[^7] Including these effects, we find the total gain of
the areal ratio and tympanic membrane will be 26.8. Using this pressure
ratio we can calculate the associated gain in decibels

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$\text{dB Gain} = 20\,log(26.8/1) = 28.6\,dB\$

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Similar to the areal ratio between the tympanic membrane and the stapes,
the areal ratio between the stapes and the oval window of the
Cochlea is 20:1

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```
$\text{dB Gain} = 20\,log(20/1) = 26\,dB\$

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```
Which yields a pressure gain of 26 dB.[^8]

### Ossicular Chain

The ossicular chain functions like a basic lever. The manubrium of the
Malleus is 1.3 times longer than
the long process of the Incus, and
the two ossicles are connect with ligaments so that they can move
together. Their attachment in the middle ear evenly distributes their
mass about an axis of rotation which allows them to be easily set into
motion by incoming acoustic signals. In addition, the ossicles are
damped enough that when the incoming signal is stopped, the malleus and
incus stop as well which is desirable as continued motion of the
ossicles would lead to echoing. The difference in lengths between the
two ossicles also corresponds to a pressure gain at the incus equivalent
to the length ratio of the two ossicles. This can be best shown by
modeling the two bones as a simple lever:

!Thumb\|center\|border\|325px\|Anatomy of the Human
Ear

Knowing this pressure ratio we are able to compute the equivalent gain
in decibels:[^9]

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```
$\text{dB Gain} = 20\,log(1.3) = 2.3\,dB\$

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```
While the pressure gain is small, it still contributes to the conduction
of sound to the inner ear.

### Overall Middle Ear Gain/ Ideal Transformer Prediction

Using the pressure gain from the ossicular chain, the areal ratio of the
tympanic membrane to the stapes, and the areal ratio of the stapes to
the oval window we find the total gain from the middle ear is

```{=html}
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```
$\text{Middle Ear dB Gain} = 2.3\,dB+ 28.6\,dB\ + 26\,dB = 56.9\,dB$

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```
This gain in the middle ear is called \"The Ideal Transformer
Prediction\...compensate\[s\] for the air-to-cochlea impedance
mismatch\" [^10] and has been confirmed in tests performed on cadavers
with the middle ears removed.

## Bone Conduction

Bone conduction is the transfer of sound to the cochlea through the
bones of the skull. In for conduction to occur in the skull, the
air-conduction threshold of sound must be exceeded by 50-60 dB which
often occur through direct vibration to the skull, primarily the
temporal bone. When this threshold is exceeded, the traveling waves
produced in the cochlea resemble those induced by the stapes.[^11] Due
to the impedance mismatch, bone conduction does not play a significant
role in hearing, but is often used for measuring hearing to determine if
there is damage to the middle ear, or to test the viability of the inner
ear.[^12]

One use for bone conduction is with people with a damaged middle ear but
having a normal functioning inner ear. For these people a traditional
hearing aid can not be used, but due to a healthy inner ear bone
conduction hearing aids can be. For these hearing aids a titanium screw
can be implanted into the skull, and an external device housing the
microphone and receiver can conduct acoustic waves to the inner ear
through the temporal bone. While hearing through bone conduction is not
perfect, it does allow for clear and understandable sound
recognition.[^13]

## References


[^1]: Gelfand, Stanley A. Hearing, an Introduction to Psychological and
    Physiological Acoustics. New York: M. Dekker, 1981. Print. p.85-102

[^2]:

[^3]: Kinsler, Lawrence E. Fundamentals of Acoustics. New York: Wiley,
    2000. Print. p.312-15

[^4]:

[^5]: Durrant, John D, and Jean H. Lovrinic. Bases of Hearing Science.
    Baltimore: Williams & Wilkins, 1977. Print.

[^6]: Auditory Function. Emanuel,Maroonroge,Letowski.
    <http://www.usaarl.army.mil/new/publications/HMD_Book09/files/Section%2016%20-%20Chapter%209%20Auditory%20Function.pdf>

[^7]:

[^8]:

[^9]: Hamill, Teri, and Lloyd L. Price. The Hearing Sciences. San Diego:
    Plural Pub, 2008. Print. p.166-69

[^10]:

[^11]:

[^12]: Yost, William A. Fundamentals of Hearing: An Introduction. San
    Diego: Academic Press, 2000. Print. p.72

[^13]:
