# Engineering Acoustics/The Human Ear and Sound Perception

**Summary:** This page will briefly overview the human auditory system,
transducer analogies, and some non linear effects pertaining to specific
characteristics of the auditory system. Results from the discipline of
Psycho-Acoustics will be
presented.

## Introduction

The human ear is a small physical device with disproportionately large
properties. On one hand it can withstand sounds with acoustic pressure
levels close to 1kPa which are pretty much the loudest encountered in
nature and on the other hand it can detect pressure levels that
correspond to displacements of the eardrum of about one tenth the
diameter of the hydrogen atom.[^1] When including the information
processing done in the brain and the physiological response that it
elicits, one can see why the Human Auditory System has been giving
researchers a hard time since the turn of the twentieth century.\
: Some researchers approached the auditory system as a very complicated,
active transducer;one that transmits the wave information first
acoustically, then mechanically, then hydro dynamically, and finally
electro-dynamically to the brain.[^2] Others like the legendary Georg
Von Bekesy, maintained that the
continuously regenerative nature of the living organism should be taken
into account when considering the behavior of the auditory system.[^3]\
: Humans however are no strangers to complicated problems. After all, we
have been to the moon, so what is going on?

## The Problem with humans

In order to explore the behavior of any physical system one needs a set
of variables describing the system.These variables should be well
defined and arise naturally from the physical principles governing the
behavior of the system.The same physical principles also provide any
researcher with well established means of assessing what constitutes a
valid measurement.\
Furthermore,in any well behaved physical system the experimenter has
control over the variables, to such an extent that he or she can hold
most of the variables constant and individually vary a few of them to
evaluate the relationship between them and quantify their dependence.\
Additionally, in any linear system the principle of superposition holds,
so that the overall effect of varying several variables at the same time
equates to a linear combination of the individual contributions observed
from varying every individual variable independently while keeping
everything else constant\
: The above mentioned usually constitute what can be described as a very
happy researcher.However, problems arise when one sets out to evaluate
the human auditory system, because hearing is a sensation and just like
every other sensation it is an esoteric process. To resolve this problem
one has to venture into the realm of psycho-physics and the principles
of psychological measurements. It is known that one cannot directly
measure sensation, but one can measure the response that the sensation
elicits.[^4]\
: With the above approach one can measure quantities such as just
noticeable differences; perceptible excitation; increased nervous
activity etc. However, the validity or relevance of those measurements
cannot readily be confirmed by first principles.[^5] The nature of the
human auditory system is such that one is not able to decouple and
independently vary any of the variables of interest(however those might
be defined) and even if one could, the principle of superposition,in
general, does not apply.

## Non Linearity \| Part1

After acknowledging the difficulties involved in quantifying the
behavior of the auditory system and developing models of hearing one
should take a look into specific sources of non-linearity and the
mechanisms through such a behavior is imposed upon the auditory system.
There is probably no better example of such behavior than what is called
the acoustic, auditory or intra-aural reflex.

### The Acoustic Reflex

The acoustic reflex in man refers to the tendency of the middle ear
muscles controlling the behavior of the ossicles (the little bones in
the middle ear) to tense under an intense acoustic stimulus, thereby
making the inner ear stiffer and in that way limiting the motion of the
stapes (the last bone in the chain). This reduction in the motion of the
stapes equates to a real rather than a perceived reduction in the
amplitude of the vibrations transmitted through the middle ear to the
inner ear. The reflex serves to protect the sensitive inner ear from
damage during exposure to loud sounds.

Unfortunately, although fast, the auditory reflex it is not an
instantaneous reaction. For low frequencies, the response takes from 20
to 40ms to be elicited and therefore offers no protection against loud
impulsive sounds like gunshots and explosions.[^6] With the onset of the
auditory reflex the entire ear exhibits a marked change in acoustic
impedance which was observed by 1934 Geffcken and measured by Bekesy and
other researchers in subsequent years. It is argued, however, that the
onset of the auditory reflex happens for sound of very high intensity
and therefore its effect on perception is limited.[^7] On the other
hand, the same reflex can be voluntarily elicited by, for example,
vocalizing. According to Lawrence A. Kinsler, it seems that the
mechanical characteristics of the ear are mainly responsible for the
response elicited by the auditory system and hence sound perception.[^8]
Whatever the exact nature of the auditory reflex may be, or what the
precise range that it has the most effect however is beyond the scope of
this article.

### Perceived loudness of pure tones

The intensity and loudness of sound are two highly interdependent
quantities. Loudness belongs to the psychological attributes of sound
and intensity is a precisely defined and measurable physical quantity.
Because of their strong similarity, the two quantities were once thought
to be one and the same, since if one increases the intensity of a
particular sound, the sound becomes louder.[^9] In the simplest and
clearest terms: intensity is measured sound level and loudness is
perceived sound level.

The measured sound level is expressed in terms of **intensity** and
**intensity level**, while the perceived sound level in expressed in
terms of **loudness** and **loudness level**.

Sound Intensity is defined as the acoustic power per unit area and it is
measured in Watts per square meter\
$$I=\frac{Power}{Unit Area}\quad\left[\frac{W}{m^2}\right]$$

However, the human ear is capable of detecting sound intensity ranging
from 1x10^−12^Wm^−2^ to 1x10^2^Wm^−2^(above which intensity permanent
damage to the ear will occur). This gives a scale in which the maximum
value is 10 000 000 000 000 times larger than the smaller one.[^10]

In order to provide more insight and get around the cumbersome numbers
we use the **Intensity level** **I~L~** which is defined as the
intensity relative to 10x10^−12^ Wm^−2^, on a logarithmic scale and it
has unit of decibels.\
.

$$I_L=10\log\left(\frac{I}{I_{ref}}\right)\quad[dB]$$

In planar waves in air and standard temperature and mean pressure,
acoustic pressure and Intensity are linked by the following relation:

$$I=\frac{P^2}{\rho c}$$

Where ρ is the air density and c the speed of sound in air. By doing the
following:\
$$I_L=10\log\frac{\left(\frac{P^2}{\rho c}\right)}{\left(\frac{P_{ref}^2}{\rho c}\right)}=10\log{\left(\frac{P}{P_{ref}}\right)}^2=20\log\frac{P}{P_{ref}}=SPL$$

The expression on the right is deemed the **Sound Pressure Level** and
it is identical to the Intensity Level, but in terms of acoustic
pressure.The reference pressure used is 20μPa. It is very close to the
average minimum audible acoustic pressure in air in the absence of any
noise.[^11] It is important to note that the minimum audible pressure is
averaged over multiple subjects, therefore for a given percentage of the
population, negative Sound pressure levels are perceptible i.e. they can
perceive sound pressures smaller than the reference pressure. The chosen
reference pressure level corresponds to the reference Intensity through
the aforementioned relationship, in a way that SPL and I~L~ are
identical.

The qualitative expressions of loud, not very loud, extremely loud, are
used to describe loudness. Although these expressions are adequate in
describing the sensation in any specific individual, they do a very poor
job in quantifying the result. The above qualitative expressions have
been made qualitative for pure tones, i.e. sinusoidal waves, with the
use of **Loudness Level** and **Loudness**.

The **loudness level** of a particular test tone is an indirect measure
of loudness and it is defined as the **Sound Pressure Level(SPL)** of a
1000 Hz pure tone that sounds as loud as the test tone.[^12] The 1000 Hz
tone was chosen arbitrarily and retained as the standard. The Loudness
level is measured in phons. The Loudness Level of the just audible
1000 Hz tone is defined as 3 phons because the minimum perceptible SPL
of a 1 kHz tone is 3 dB. Increments in phons are logarithmic because the
SPL is measured in decibels.

The **loudness level** is very useful in quantifying the sensation,
however it fails to provide information on the relation between sounds
of different loudness levels. In other words it fails to provide insight
on how much louder a sound of e.g. 20 phons is than a sound of 50 phons.
To get around this problem, we use **Loudness** which has units of
sones. **Loudness** is based on the 40 dB, 1000 Hz pure tone which is
defined as to have a loudness of 1 sones. The Loudness scale is derived
by increasing or decreasing the SPL of the 1 kHz tone until it \"sounds
twice as loud as before\" or \"half as loud \" etc. Successive halving
of the loudness creates the rest of the scale. The **Loudness** for the
remaining tones is determined by the same equal loudness judgment that
provides the **Loudness Levels**.[^13]

Loudness and Loudness level are best illustrated and are most useful
when plotted against the SPL of pure tones, in what are called equal
loudness contours or Fletcher & Munson
curves, so named after the
earlier researchers, however the way loudness is measured has been
significantly altered and standardized since the time when such
measurements were first made.\

<File:Lindos1.svg%7CEqual> Loudness contours

## A little bit of bio

### The cochlea

The **cochlea**, or inner ear, constitutes the hydrodynamic part of the
ear. It is a small, hollow, snail shaped member formed from bone and
filled with colorless liquid. It has an uncoiled length of about 35mm
and a cross-sectional area of about 4mm^2^ on the end closest to the
inner ear, that tapers off to about 1mm^2^ at the far end.[^14]\
: It is filled with **two** different fluids separated in **three**
different channels that run together from the base of the stapes to the
apex of the cochlea, however two of the channels are separated by
Raleigh\'s membrane, which is thin and flexible enough to be neglected
from a hydromechanical point of view.[^15] The vibrations are
transmitted directly from the base-plate of the stapes, the last of the
three ossicles, to the contained fluid. The cochlea is divided down the
middle by the **basilar membrane** which is a partly bony and partly
gelatinous membrane. It is on this membrane that the organ of corti and
the infamous **hair cells** reside.

<File:Anatomy> of Human Ear with Cochlear Frequency Mapping.svg\|The
Auditory system and Cochlea <File:Cochlea.png%7CThe> three Fluid filled
cavities

### The basilar membrane

As previously mentioned, the basilar membrane is a flexible gelatinous
membrane that divides the cochlea longitudinally. It is the flexible
part of the cochlear partition (the other being rather bony)and it
contains about 25 000 nerve endings attached to numerous haircells
arranaged on the surface of the membrane. It extends from the base to
just before the apex of the cochlea at which point it terminates at the
helicotrema. This creates two hydromechanically distinct channels, with
the baseplate of the stapes attached to the entrance of the upper
channel at the **oval window**, and a highly flexible membrane called
the **round window** sealing the lower channel. The two channels connect
at the apex through the **helicotrema** which is basically a gap through
the cochlear partition.\
:

<File:Gray928.png%7CDiagrammatic> longitudinal section of the cochlea
showing the location of the Basilar Membrane <File:Two> Views of
Cochlear Mechanics.svg\|Two Views of Cochlear Mechanics

:   

The vibrations transmitted to the stapes set up acoustic waves in the
fluid that travel down the upper channel, through the helicotrema and
back up through the lower channel. Since the walls of the cochlea are
relatively rigid and the contained fluids relatively incompressible,
this causes the basilar membrane to flex. In order to equalize the
pressure in the cochlea, the round window \"bulges out\" and in this way
provides pressure release.

:   

The basilar membrane starts out narrowly, with a width of about 0.04mm
near the oval window and then widens to about 0.5mm near the
helicotrema. This non uniformity in width, along with the pressure
release provided at the round window cause the basilar membrane to
exhibit maxima of vibration at different locations (distances from the
oval window) along the membrane, depending on the frequency of
vibration. This makes the basilar membrane behave as an acoustic filter
that separates the constituent frequencies of an incoming sound signal
depending on the location of the maxima.\
:Basilar Membrane
Animation

:   

<File:Uncoiled> cochlea with basilar membrane.png\|Uncoiled cochlea with
basilar membrane

:   

### Whats up with all the hair?

The hair-cells that populate the top surface of the basilar membrane are
the last part in the chain of transformation of the mechanical energy of
the acoustics wave into electrical impulses. These cells are arranged in
an inner row and an outer row in the organ of corti (which runs along
the basilar membrane) and they are surrounded by electrically charged
cells at different potentials(synapses).[^16][^17]\
:

<File:Gray932.png%7CSection> exposing the hair and hair-cells
<File:Cochlea-crosssection.svg%7CCochlea-crosssection> with the
hair-cells visible

:   

As was already mentioned, the basilar membrane exhibits various
vibration maxima at different locations when excited by a sound input.
As a result of these excitations a relative motion of the fluid parallel
to the membrane is effected. This motion produces a shear force on
myriads of minuscule hairs protruding from these cells. The disruption
produces an electrochemical cascade on the surrounding electrically
active cells, which results in a signal to be sent to the brain.\
: What is really important is to note that these hair-cells are not
evenly distributed over the surface of the basilar membrane, but rather
they are concentrated in discreet patches. Since different frequencies
make different different parts of the membrane vibrate more than others,
this means that there are ranges of frequencies that we can perceive
better than others, depending on the number density of the hair-cells
surrounding the corresponding region on the basilar membrane. This
introduces discreetness and gives a sort of minimum resolution to our
sense of hearing, thus causing some interesting non linear effects to be
discussed soon.

:   

<File:Hair> Cell Patterning Defects in the Cochlea.png\|Arrangement of
the hair cells on the cochlea. Left=healthy;Right=pattern defects

:   

Due to the similarity between the behavior of the inner ear and the
behavior of a band-pass filter, the above groups of frequencies have
been named **critical bandwidths**.[^18]

## Non Linearity \| Part 2

Now that a little bit more has been presented about the workings of the
inner ear, more peculiarities of the idiosyncratic auditory system can
be illustrated, starting with a non linear effect that is fairly common
and very noticeable when it occurs. It is the phenomenon of beating.

### Beating Phenomena

Beating phenomena are a characteristic of multiple degree of freedom
systems, where the various degrees of freedom are coupled to some extent
and that receive two harmonic excitations at slightly different
frequencies. The excitations can be summed as follows:[^19]

:   $$x=A_1e^{j(\omega_1 t)+\Phi_1}+A_2e^{j(\omega_2 t)+\Phi_2}$$

The resulting vibration is no longer simple harmonic.

:   

The inner ear is a continuous system, with the basilar membrane serving
as a complicated bandpass filter to separate frequencies. When one or
both ears are exposed to sound that consists of two tones with a slight
difference in their frequencies, the non uniform distribution and strong
localization of the hair cells on the surface of the basilar membrane
result in the same group (or critical bandwidth) of hair cells being
excited by both tonal components of the incident sound.\
:

<File:Beating> Frequency.svg\|Beating

As a result, the listener perceives the combination sound as that of a
single frequency tone but with periodically varying intensity. This is
known as beating.\
: The tones remain indistinguishable until the frequency separation
between them , is greater than the a bandwidth. It is really interesting
to note that if the two tones are presented to each ear separately, then
no beating occurs and the ear is able to resolve the difference.[^20]

## References

## Links and Resources

-   Dancing Hair Cell
-   American Society of Acoustics
-   Table of sound
    levels
-   Basilar Membrane
    Animation

[^1]: Acoustics, Leo L. Beranek 1993, Copyright: Acoustical Society of
    America, Ch13 Hearing, Speech Inteligibility and Psychoacoustic
    Criteria

[^2]: Overall Loudness of Steady Sounds According to Theory and
    Experiment, Walton L. Howes, Nasa reference Edition 1001

[^3]: THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA VOLUME 23.NUMBER
    5, SEPTEMBER.1951,DC Potentials and Energy Balance of the Cochlear
    Partition,GEORG V. Bekesy,Psycho-Acousic Laboratory, Harvard
    University, Cambridge, Massachusetts,(Received May 5, 1951)

[^4]: The Measurement of Hearing,Ira J. Hirsh, McGraw Hill Book Company,
    Inc, First Edition,1952

[^5]:

[^6]: Fundamentals of Acoustics,Lawrence E. Kinsler, Alan B.
    Coppens,4rth Edition

[^7]: Acoustic Reflex in Man,Aage R. Moller, J. Acoust. Soc. Am. 34,
    1524 (1962), <DOI:10.1121/1.1918384>

[^8]:

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[^10]: <http://www.engineeringtoolbox.com/sound-intensity-d_712.html>

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