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Hearing: Ear work and place Theory of hearing, Lecture notes of Advanced Physics

Hearing in breifly explain place theory of hearing, physical sound intensity, logarithms, fundamental tracking and aural harmonics.

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7 – Hearing
Human ear is, probably, the most
remarkable organ. It has a very
sophysticated construction, is
sensitive to the sound of frequency
from 20 Hz to 20 KHz, that is, in the
range of three decades in frequency,
and of intensity from 10−12 W/m2to 1
W/m2, that is, twelve decades in the
intensity! Ears of some animals are
even better.
Human ear (schematic)
Human ear (realistic)
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7 – Hearing

Human ear is, probably, the mostremarkable organ. It has a verysophysticated construction, issensitive to the sound of frequencyfrom 20 Hz to 20 KHz, that is, in therange of three decades in frequency,and of intensity from

W/m

2

to 1

W/m

2 , that is, twelve decades in the

intensity! Ears of some animals areeven better.

Human ear (schematic)Human ear (realistic)

How the ear works

  • The sound enters the auditory canal from the outside and reaches the eardrum• The eardrum vibrates and sets the three ossicles, hammer, anvil, and stirrup in motion• The stirrup pushes at the oval window in the cochlea and produces waves in its liquid• The waves propagate in the scala vestibuli along the basilar membrane, reach the apex of thecochlea, enter through the helicotrema into the scala tympani and propagate back in the scalatympani along the other side of the basilar membrane• Waves reach the round window and are being damped there• The travelling waves in the cochlear liquid excite the nerve endings located at the basilarmembrane, and the signals go to the brain that decodes them as signals of particular frequencyand particular loudness. The length of the basilar membrane is about 3.5 cm and it contains about35000 nerve endings called „hair cells“.• In fact, the magnification of the region of the basilar membrane shows another two membranes,and the whole process is more complicated

Base

Apex

Place theory of hearing

It was proven experimentally that thesound of a particular frequency createswaves in the cocklear liquid that haveantinodes at particular well-definedplaces of the basilar membrane alongits length. That is, each particularfrequency excites the hair cells at aparticular distance from the base of thebasilar membrane. The brain knowsfrom where the nerve signals arecoming and decodes the positionsalong the basilar membrane into thesensation of frequency. It was shownthat equal distances between thesensitive points for any two soundscorrespond to equal ratios of thefrequencies of the two sounds. That is,the frequency coding is logarithmicalthat is the explanation of a largefrequency range the ear is sensitive to.

Here the frequencies double each time, and this resultsto the same shift down along the vertical axis

Hair cells

Sound intensity response of the ear

The response of the ear to the sound intensity

is even more remarkable than its frequency

response, so that we can hear both very quiet and very loud sounds. The intensity response ofthe ear is logarithmic

, too. Below the sound-intensity level

(SIL) in decibel

is shown on the left and

the physical sound intensity in W/m

2

is shown on the right. The curves are the sould intensity

levels that are percepted by an average person with normal hearing as equally loud.

Logarithms

0

20

40

60

80

100

(^210) - log

10 ) x (

x

2

10

log

100

log

, 1

10

log

, 0

1

log:

Examples

log

then

,

10

Let

2

10

10

10

10

10

=

=

=

=

=

=

y

x

y

x

)

log(

)

log(

1

log

)

log(

)

log(

)

log(

)

log(

)

log(

1

a

a

a

a

a

b

a

ab

=  

 

=

=

α

α

Properties: Particular values:

0

2000

4000

6000

8000

10000

1.00E+0088.00E+0076.00E+0074.00E+0072.00E+0070.00E+

x

y

=

x

2

Standard plot of a quadratic function. Drawback: The region of small

x

is not well represented

1

10

100

1000

10000

0.1 0. 10 1 1000100 10000010000 1000000

1E8 1E

x

y

=

x

2

Double logarithmic (log-log) plot of the same function. All ranges of

x

, small and large, are

equally well represented. The ear was constructed to hear both very quiet and very loudsounds of very small and very large frequencies. This is why the ear hears logarithmically.

Aural harmonics Fundamental tracking

The response of the ear is nonlinear

, so that loud sounds get distorted in the ear. This phenomenon

is similar to the „clipping“ in electric circuits. As a result, the signal remains periodic but it is nolonger pure sinusoidal. Its Fourier spectrum contains harmonics of the main tone. Since theseharmonics are produced by the ear, they are called aural harmonics

Combinational tones

In the case of two incoming signals with frequencies

f

1

and

f

, nonlinearities in the ear result in the 2

appearance in the Fourier spectrum of the signal (in the ear!) of many combinational tones withfrequencies

tones)

e

(differenc

tones)

(sum

2

1

2

1

nf

mf

f

nf

mf

f

where

m

and

n

are natural numbers. The combinational tones are weaker than the main tones but

they can be made apparent by adding a weak tone with the frequency close to one of thecombinational tones to produce beats.Our brain does an additional job of (mis)interpreting the incoming sound wave. In the case of twosounds with frequencies that can be different harmonics of one fundamental, our brain adds thisnonexistent fundamental to our perception of the sound. For instance,

f

=500 Hz and 1

f

=700 Hz, we 2

also hear the „fundamental“ 100 Hz. The same happens for

f

=600 Hz and 1

f

=700 Hz, of course. 2

But for

f

=550 Hz and 1

f

=700 Hz, we do not hear this „fundamental“. 2

200

400

600

800

1000

1200

1400

20 15 10 5

Fourier spectrum of the signal consisting of the two pure tones

f

=500 Hz and 1

f

=700 Hz distorted 2

by a nonlinearity in the ear (or in the speakers!)

Main tones

Difference

tone

Sumtone

Second

aural harmonic

Second

aural harmonic