Download The harmonic hydro-mechanical movement of the

Buenos Aires – 5 to 9 September, 2016
st
Acoustics for the 21 Century…
PROCEEDINGS of the 22
nd
International Congress on Acoustics
Psychological and Physiological Acoustics (others): Paper ICA2016-288
The harmonic hydro-mechanical movement of the
cochlear fluid is not a wave neither a vibration
Santos Tieso(a), Lucas Fantini(a), Francisco Messina(a), Nahuel Cacavelos(a), Gilda Farelli(a),
Leonardo Zavala(a), Maria Tieso(a), Sebastian Iezzi(a), Federico Adrián Bosio(a).
(a)
Universidad Nacional de Tres de Febrero, Argentina,[email protected]
Abstract
Physical descriptions of current harmonic movements are inadequate to describe the
phenomena occurring within the cochlear physiology. In this paper the hydro -mechanical
harmonic movement of the cochlear fluid is described. It is possible to distinguish this type of
movement as a different physic category of a vibrating solid in a mass–spring–damper system
and the wave propagation in a medium. Also, it is conceptually explains similarities and
differences between the three types of movement mentioned. Finally, it highlights the
importance of the existence of the movement in the volume perceived relating it with the
different phenomena occurring in the ear.
Keywords: inner ear; cochlea; hydro-mechanical movement; biomedical
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22 International Congress on Acoustics, ICA 2016
Buenos Aires – 5 to 9 September, 2016
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Acoustics for the 21 Century…
The harmonic hydro-mechanical movement of the
cochlear fluid is not a wave neither a vibration
1 Introduction
It’s essential for the study of the cochlear physiology to have a deep understanding of the
harmonic hydro-mechanical motion. This motion cannot be framed in terms of vibration, wave or
a piece of wave in the proximities of the source. This kind of motion should be framed in a
different type of sinusoidal movement defined as the result of the heterogeneous pressure field
transferred to the harmonic hydro-mechanical system´s fluid. To simplify the understanding of
this motion, the simple harmonic hydro-mechanical system (S.H.H.S.) is defined. It’s shaped as
a cubic recipient with two circular windows with equal areas. These two windows, called oval
and round windows, are located in opposite walls of the recipient, like shown the Figure 1. The
distance between both windows should be much smaller than the wavelength of the stimulus
applied to the windows [1].
The laws of physics that apply to the motion of the fluid in this system are the same that apply
for physiological behaviour of the cochlea. These laws determine the different signs and clinical
symptoms of the auditory related pathologies.
Source: (Tieso, 2013)
Figure 1: Two-dimensional representation of a S.H.H.S. OW is Oval Window and RW is Round
Window.
.
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22 International Congress on Acoustics, ICA 2016
Buenos Aires – 5 to 9 September, 2016
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Acoustics for the 21 Century…
2 Theoretical development
The H.H.M. depends on two different impedances:
•
Hydro-mechanical impedance
•
Particular impedance
The differential harmonic pressure applied to the windows of a S.H.H.S is used in generating
volumetric velocities (Qh) over the resistance, volumetric accelerations to the inertance and
volumetric elongation to the elasticities. From this, the general expression of the hydromechanical impedance can be derived.
𝑍! = 𝑅! + 𝑗 𝑚! −
𝑘!
𝜔
(1)
Where
•
𝑍! is the hydro-mechanical impedance,
•
𝑅! the hydro-mechanical resistance (provided by the windows),
•
𝑚! the hydro-mechanical inertance,
•
𝑘! the hydro-mechanical elasticity (provided by the windows),
•
𝜔 the angular frequency.
Its definition is equivalent to
𝑍! =
Δ𝑝
𝑄!
(2)
Where
•
𝛥p the pressure difference applied to the system and
•
Q the hydro-mechanical flow.
h
Special considerations should apply to the hydro-mechanical stimulus. It can’t be considered as
the pressure applied to the oval window. The information about the pressure applied to one
window doesn’t guarantee the existence of an hydro-mechanical flow between windows nor
does the absence of pressure in one window discards it. The application of an harmonic
pressure to one windows only indicates that an harmonic pressure field has been introduced to
the inner fluid, but does not specify the characteristics of the mentioned pressure field. The
magnitude of the pressure difference between windows in relation to the systems own
resonance and its quality factor are responsible for the flow quantity [2].
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22 International Congress on Acoustics, ICA 2016
Buenos Aires – 5 to 9 September, 2016
st
Acoustics for the 21 Century…
The characteristics of the harmonic pressure field, transferred to the fluid in a harmonic hydromechanical system (H.H.S), depend on five factors:
1. The applied harmonic pressure module
2. The phase differences
3. The stimulus frequencies
4. The own systems resonance frequencies
5. The quality factor
In the particular case of application of the stimulus to one window, pressure magnitude is equal
to the pressure difference that affects the whole system. If the described S.H.H.S lacked of its
round window, it could not be considered an harmonic hydro-mechanical system, because any
harmonic pressure applied to the oval window could only transfer a homogeneous harmonic
pressure field to a liquid fluid, without existence of pressure differences and, consequently, with
no gradient responsible of inertial movement. The Figure 2 shows this case as opposed to a
S.H.H.S. with the two windows.
Source: (Tieso, 2013)
Figure 2: Two-dimensional representation of: (a) a S.H.H.S. without the round window. POW is the
harmonic pressure applied to the oval window and PL is the homogeneous harmonic pressure
field transferred to the liquid; (b) a S.H.H.S. in which a pressure is applied in each window. PRW is
the harmonic pressure applied to the round window. The flow is the result of relationship of both
and the S.H.H.S.
In any harmonic hydro-mechanical system, due to its “mass-spring-damper” series kind,
differential pressures are distributed for a common flow [4]. Any simple hydro-mechanical
system, stimulated in its resonance frequency, responds so that its flow is in phase with the
stimulus differential pressures. But if such system is stimulated with a frequency greater than its
resonance frequency, its response is delayed in phase. On the other hand, if the referred
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22 International Congress on Acoustics, ICA 2016
Buenos Aires – 5 to 9 September, 2016
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Acoustics for the 21 Century…
system is stimulated with a frequency lower than its resonance, the responses phase overtake
the differential pressures phase. This is shown in Figure 3 [4] [5].
Source(www.mathworks.com, 2016)
Figure 3: Phase vs normalized frequency between flow and differential pressures on a S.H.H.S.
A general expression for the particular impedance, corresponding to the acoustic particles in the
fluid of the S.H.H.S, arises from the application of Euler’s equation to the harmonic movement
[6]:
−∇𝑝 = 𝛿𝑎! (3)
𝑎! = 𝑗𝜔𝑣! (4)
If the stimulus is harmonic then
Then, recalling the definition of particular impedance [1], the following relation is obtained:
𝑍! = −
∇𝑝
= 𝑗𝛿𝜔
𝑣!
(5)
The liquid fluid in any S.H.H.S. behaves as an inertance, which is the inelastic and non-resistive
component of the system. In the vibratory movement, a force accelerates the mass of the
system, which is a density with volume, but in the H.H.M, the differential pressure accelerates
an inertance.
In a vibrational system the mass determines the harmonic linear inertial movement, which is
independent of the mass geometry. However, in a harmonic hydromechanical system, the mass
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22 International Congress on Acoustics, ICA 2016
Buenos Aires – 5 to 9 September, 2016
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Acoustics for the 21 Century…
is not fundamental, because the same mass value in different geometries result in different
systems inertances, and different mass values with same geometry could be related to the
same inertance.
In the vibratory movement of a solid body, the concept of acoustic particle is diluted, provided
that the stimulus frequency is such that the solid is moved as a whole [6]. If the value of the
stimulus frequency is too high, a wave could be introduced in the system. In the H.H.M. the
acoustic particle is still valid, but it is no part of propagation of any wave.
This movement cannot, and should not, be confused with a vibration of liquid fluid. In any
vibrational movement of a solid, the force determines the velocity. But in the liquid, the pressure
difference is the responsible.
The behavior of acoustic particles in a H.H.S. is similar to Euler’s fluid particle for the study of
inharmonic laminar flows, due to its inertial-type speed, but particularly in a H.H.S. the particle
speed is harmonic, or sinusoidal. It’s this harmonic movement which gives Euler’s particle such
an acoustical quality. The divergence of the acoustic particle velocity is zero for all H.H.S.
Moreover, the behavior of the acoustic particle in a H.H.S. is different from its behavior in a
plane wave propagation. In the propagation of the plane wave in an isotropic, homogeneous
and infinite media, the particle velocity is mainly active, and completely divergent, i.e. the
acoustic particle velocity is given with density changes [6], but in the H.H.M. the velocity does
not need any density changes to appear.
In the wave propagation, pressure is related to velocity through the characteristic impedance:
𝑍! =
𝑝
𝑣!
(6)
In this case, p is the air pressure and v is the particle velocity in the plane wave propagation
environment.
p
Specifically, in this type of propagation, pressure and velocity have the same phase, and the
impedance does not depend on the stimulus frequency, in fact, it is equal to the characteristic
impedance of the environment.
In the liquid of a H.H.S. the acoustic particle velocity depends on the pressure gradient, and the
particular impedance offered to the mentioned pressure gradient. This relationship from
Equation (5) the following relation is obtained:
𝑣! = −
∇𝑝
𝑍!
(7)
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22 International Congress on Acoustics, ICA 2016
Buenos Aires – 5 to 9 September, 2016
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Acoustics for the 21 Century…
Hence, the particle velocity also depends on the angular frequency of the stimulus, as shown in
Equation (5).
In the H.H.M there is no univocal relationship between pressure applied to a particle and its
velocity as in the plane wave propagation, thereby, quantifying pressure is not enough to find
particle velocity, even at the same frequency.
The differential pressure stimulus is applied in phase to every point in the liquid, however the
pressure gradient that affects different acoustic particles can be given with phase differences, if
the harmonic hydromechanical system is not simple.
In the harmonic inertial movement of the liquid in any H.H.S. the acoustic particles oscillate at
the stimulus frequency. The oscillatory movement of all particles in the H.H.S. determines the
geometry of the inertial movement or flow, which is the inertance geometry.
The H.H.M. is easily distinguishable from the wave propagation, because a wave can be
represented by a function of space and time, but the H.H.M. is a function only of time.
In the spherical wave propagation, considering a very small kr, the particle velocity is principally
inertial, but it’s still depending on the pressure that affects it, while in the H.H.M. the particle
velocity depends on the pressure gradient, transferred to it by the stimulus. Hence, it is possible
to find in a H.H.S acoustic particles that are affected by small pressures with high speed, and
vice versa, while in the spherical wave propagation with small kr, greater pressures cause
necessarily greater velocities.
In a H.H.S. it is possible to find acoustic particles subjected to equal pressures but moving with
different velocities, while in the spherical wave propagation in the vicinity of the source, equal
pressure implies equal particle velocity.
3 Conclusions
In the present paper a new type of motion was proposed to describe the behavior of the
cochlea's inner fluid. This system, called hydro-mechanical harmonic, has new features and
whose differences with a vibrating system and with an acoustic system were detailed to mark
some of the conceptual differences between those systems. Through mathematical and
physical development it was concluded that this type of motion while being of an harmonic
nature can’t be described as a wave nor a vibration. This approach shines a new light on which
to look and study the inner ear and it’s related pathologies. The basic concepts of the resonant
motion, as well as an acoustic system, do not provide answers to all existing phenomena in the
cochlea. Starting with the different nature of the stimulus, and its relationship with different
impedances, we can see that the way to respond to that stimulus in the three systems are
essentially different. Given the conditions of the inner ear physiology, particularly its component
liquid and the relationship with the windows and the dimensions of cochlea, the harmonic
hydromechanical system responds more adequately to the behavior of inner ear manner.
In the vibrating system, the estimated involves a transfer of force on a mass and acceleration,
and in the acoustic system, the movement depends on a variation in the density of molecules.
While, the hydro-mechanical system not responds to a force applied to a mass but rather, it
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Buenos Aires – 5 to 9 September, 2016
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Acoustics for the 21 Century…
does responds to a pressure gradient as a stimulus applies to a density geometry. Nor does not
suffer change in the fluid density that composes nor has dimensions to develop a plane wave,
with its respective length.
The analysis of the impedances involved in the new system described gives us the idea of the
nature of this movement and its difference with the other mentioned system. Several studies
can be performed in the future in order to enlarge the knowledge about the cochlea’s inner fluid
motion. Scale model and digital simulations of the S.H.H.S. can be designed and compare to
the theoretical development. For a more complex work, a live cochlea can be measured using a
laser vibrometer. This way the velocities of the inner cochlea particles can be measured.
References
[1] Tieso S. Fisiologia auditiva. Corrales ediciones médicas, Buenos Aires (Argentina) 2006.
[2] Geisler D. From sound to synapse, Oxford University Press, USA (1998).
[3] White F. Fluid Mechanics, McGraw Hill, 1979, pp 37-39.
[4] Recuero M., Ingeniería acústica, Editorial Paraninfo S.A, 1995.
[5] Robert L. Boylestad, Introductory Circuit Analysis, Prentice Hall, 2010.
[6] Claudio Mataix, Mecánica de Fluidos y Máquinas Hidráulicas, Alfaomega 2nd edition, 1986.
[7] Rossing, T. Springer, Handbook of Acoustics. Springer, NY (USA), 2nd edition, 2015.
[8] Tieso S. Fundamentos de bioacústica, Eduntref, Buenos Aires (Argentina), 2013.
[9] Beranek, L. Acoustics. Springer, NY (USA), 2nd edition, 2004.
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