164
D. Rocchesso: Sound Processing
impedance of the tube
acoustic intensity
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Pinna
Ear canal
Eardrum
Oval window
Round window
Tectorial Membrane
Basilar Mrmbrane
Hair Cells
Acoustic Nerve
Scala Vestibuli
Scala Tympani
Helicotrema
Base
Apex
Outer ear
Middle ear
Inner ear
Figure 1: Cartoon physiology of the ear
membrane oscillates and transversal waves are propagated. The basilar mem-
brane can be thought of as a string having a decreasing tension as we move from
the base to the apex. This tension changes by about four orders of magnitude
from base to apex. Along a string, the waves propagate at speed
c =
T
L
=
Tension
Linear density
,
(1)
and the wavelength associated with the component at frequency f is
=
1
f
T
L
=
c
f
.
(2)
The impedance of the tube is z
0
=
L
T and, if v
max
is the peak value of
transversal velocity, the wave power is
P =
1
2
z
0
v
2
max
=
1
2
L
T v
2
max
.
(3)
While a wave component at frequency f is propagating from the base to the
apex, its wavelength decreases (because tension decreases) and, due to the physi-
cal requirement of power constancy, its amplitude increases. However, this prop-
agation is not lossless, and dissipation increases with the amplitude, so that a
frequency-dependent maximum region will emerge along the basilar membrane
(see figure 2). Since the high frequencies are more affected by propagation losses,
their characteristic resonance areas are cluttered close to the base, while low fre-
quencies are more widely distributed toward the apex. About two thirds of the
length of the cochlea is devoted to low frequencies (about one fourth of the
audio bandwidth), thus giving more frequency resolution to the slowly-varying
components.
C.2
Sound Intensity
Consider a sinusoidal point source in free space. It generates spherical pressure
waves that carry energy. The acoustic intensity is the power by unit surface that