context: This question may appear to be off topic because it is about the biological particulars of the human ear. Physics can solve abstract models inspired by living organisms, but it is biology that is tasked with developing those models.
Maybe I can answer my own question, preceded by a number of basic principles in physics, ( detailed mathematical explanation and analytical calculation, been excluded to give in all details, – more than likely difficult comprehend due to the complexity of the subject matter, which is pure physics - ), and that indeed applying the Bernoulli effect is correct when in an oscillating column of fluid all conditions for a quasi-stationary potential flow are met.
Such an quasi-stationary potential flow will not only lead to the valid use of the Bernoulli effect, but will also perfectly fulfil Laplace‟s equation, which, of course, is another prerequisite.
Consequently, in the case we do not have to make use of Navier-Stokes equations at all.
Preceded by taking data found in literature as our point of departure; namely, that for normal sound stimuli the maximal displacements of the oval window are substantially smaller than 10 micrometer.
Since the perilymph can be regarded to be moving as a whole, we can calculate the velocity of the perilymph anywhere along the basilar membrane as a time derivative. The velocity amplitude in the perilymph will then become. Thereupon, we can make use of data found in standard literature for a number of parameters that will play a role in this process:
a. For the perilymph duct in the cochlea, a given: channel diameter mm.
b. While the kinematic viscosity is given by: m2/s.
c. Perilymph is regarded as a practically incompressible fluid, similar to water.
If, using these data with their values, we calculate for the Reynold‟s number we arrive at. Which is 1/5 of the critical value required to change the flow conditions from laminar, for low, to turbulent for high.
As any physicist with some knowledge of this subject knows, the critical point is found between 2000 and 3000. The flow of the incompressible perilymph fluid thus remains laminar for all sound frequencies in the cochlea. It follows, that the involved parameters remain unchanged for all regular sound signals that generate alternating perilymph velocities.
Therefore, the quasi-static approach may be used by replacing the constant velocity with the time dependent velocity.
For the concept, the definition of „quasi-static‟ allows us to regard the time dependent behaviour as a large number of sequential multi-moment observations, for which the factor time increases from moment to moment, in infinitesimal steps.
This is completely in accordance with the manner in which Navier-Stokes equations for laminar incompressible fluids can be reduced to the Law of Bernoulli in the four-dimensional vector representation.
Because the perilymph duct actually functions as a stream channel, we may even use the standard scalar representation for the Bernoulli effect.
For a sound signal with frequency and a deflection amplitude for the oval window, the relation between the pressure change on the wall of the perilymph duct and the perilymp velocity is given.
This quasi-static relation between pressure variations in the perilymph fluid and the sound pressure for all frequencies complies with the law of conservation of energy, resulting in a pressure signal over the entire basilar membrane.
Question:
Why is the vector sum of perilymph velocity, in the overall push-pull movements of the incompressible perilymph fluid in the combined scala vestibuli and scala tympani, create audible or not audible sound, as seen in the relation between sound stimulus and electrical output in the cochlea?
Why not audible sound in this example?
Regarding Waver & Lawrence 1950, The Acoustic Pathways to the Cochlea, JASA 22: 460, Heerens & de Ru (Applying Physics Makes Auditory Sense, PDF here) give in essence an utterly legitimate supplementary, both well defined and logically based in the experiments by Wever and Lawrence (1950) – later verified by Voss et al. (1996).
Two sound pressure signals that are equal in intensity and that simultaneously attempt to push-pull the oval window and the round window, in the same direction in and out of the cochlea – causing window movements in the same phase – will result in a vector sum for the perilymph stimuli of zero. This means that the perilymph velocity is zero as well. Wever and Lawrence established that in this case the changes in cochlear potentials are zero.
My own guess is:
- The velocity of the perilymph fluid within the cochlear duct evokes the change in cochlear potentials.
- It can easely be demonstrated with vectors that the sum of two vectors of equal modulus [length of vector], which phase changes between 0 and 180 degrees from each other, the square thereof increases from 0 to 4 times the vector length, and the phase will change from 0 to 90 degrees.
May we therefore conclude that the change in cochlear potentials is dependent on the resulting perilymph velocity and not on the pressure load?
Why audible sound in this example?
Heerens & de Ru also say
In the practical case of the Vibrant Soundbridge the floating mass transducer actually evokes the force on the round window, which in turn sets the perilymph in the cochlea into motion.
My own guess is:
Naturally, this velocity is generated by the impact of a net force on the perilymph.
The perilymph fluid in the scala vestibuli and the scala tympani moves as a whole, as it forms an incompressible fluid column.
The velocity of the perilymph fluid within the cochlear duct evokes the change in cochlear potentials.
May we therefore conclude that the change in cochlear potentials is dependent on the resulting perilymph velocity and not on the pressure load?
context
I read that the authors Heerens & de Ru clearly discuss, that there was two cochlear potential contributions that were measured by Wever and Lawrence in their experiments (1950).
My guess: When there is a quasi-static relation between pressure variations in the perilymph fluid and the sound pressure for all frequencies it complies with the law of conservation of energy.
Resulting in a pressure signal over the entire basilar membrane. As a consequence the basilar membrane reacts to the sound energy signal, equally evoked anywhere on this membrane.
The sound energy signal also evokes a DC cochlear potential signal on this membrane as a result of the constant pressure load. Furthermore, an AC cochlear potential signal proportional to the pressure load is evoked in a small area located near the place of resonance on the basilar membrane.
I read in the manuscript from Heerens & de Ru that they clearly discuss, that it was these two cochlear potential contributions that were measured by Wever and Lawrence in their experiments (1950). As both DC and AC are proportionally related to the sound pressure amplitude squared, a double value of amplitude will result in a four times higher pressure load on the basilar membrane, which equals a 6 dB increase of cochlear potentials.
Because the total pressure in the perilymph, wich is equal to the pressure on the basilar membrane, is not a function of the coordinates (x, y, z), however is only dependent on the time variable t, the solution, at any moment, satisfies La Place‟s equation: ...
context acoustics: The acoustic pathways, So I tagged it with the tag acoustics: Acoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound and infrasound. Applications of acoustics are for instance the audio and noise control industries.
context acoustics:
From diyaudio.com : www .diyaudio.com/forums/lounge/200865-sound-quality-vs-measurements-304.html#post2931925 post number: #3034
About Heerens and de Ru careful analysis of the auditory pathway experiments that were
executed by Wever and Lawrence,
a remarkable experiment in 1950:
Ernest Glen Wever and Merle Lawrence:
The acoustic pathways to the cochlea, JASA July 1950, 22: 460-467
experiment:
- They (Wever and Lawrence) removed the cat eardrum and ear-bone-chain.
- Brought in a tube around the round window.
- Stimulated with pure tones as follows:
only on oval window.
only on round window.
on both windows different in phase from 0 ° - 180 °.
- Recorded the cochlear microphonics, [CM]
- signal which corresponds to signal to the brains.
Results in CM changes:
- Windows separately with the same signal: CM: equal changes
- Both windows simultaneously in the same direction: CM: no change
- Both windows in the opposite direction [180 ° or pi]: CM: maximum change
- Maximum is 6 dB higher than the two stimuli separately. (Recorded by Wever
and Lawrence)
Conclusions Wever & Lawrence:
- Both paths provide identical signal in auditory nerve.
- Over much of the frequency range:
Oscillations in phase on oval and round window: minimum
Vibrations in antiphase on oval and round window: maximum
Similar research Voss, Rosowski, Peake (1996):
- Differential pressure oval - round window: signal.
- Signal Components: DC [DC] AC [AC]
Heerens and de Ru conclusions:
- Signal to arise in the brains by: moving the perilymph.
- Two identical stimuli moving opposite, (as in the experiments that were executed by
Wever and Lawrence ), supplies total movement: 2 times as large.
But electric signal is not: 2 but 4 times as large.
6 dB (10.log 4 = 6.0).
For: 6 dB = 10 × 10log 4
Yes we are talking about "change in potential".
And we're not talking about the "ever-present potential".
It is namely the change of the present potential, which increases by a factor of 4,
if the perilymph speed increases by a factor of 2.
Still according to the mathematical relationship that 6 dB = 10 × 10log 4.
This results in:
The signal generated in the auditory nerve is proportional to the square of the perilymph
velocity.
and to complete the context:
The varying pressure in the outer ear canal evoked by sound stimuli activates the eardrum. The eardrum, in turn, brings the ossicular chain into motion. This motion is transferred via the stapes, which is closely connected to the oval window, to the perilymph within the cochlea.
Entirely consistent with the findings of Wever and Lawrence in 1950, it is argued that direct sound stimuli on either the oval window or on the round window, the other membrane separating the middle ear from the cochlea, cause a similar change in the electrical cochlear potential related to the perilymph motion.
But there is more to say about the findings of Wever and Lawrence.
In their experiment they stimulated in different ways oval and/or round window of a cat’s cochlea and measured the corresponding cochlear potentials. The results of these experiments as they have reported are:
- For all the different frequencies spread over the entire audible spectrum, used for the identical stimulus of either the oval or the round window, it was found that both stimuli resulted in identical changes in cochlear potentials.
- Simultaneous stimulation of both windows in phase – i.e. both windows stimulated to move in and out of the cochlea – resulted in zero change in cochlear potentials.
- Simultaneous stimulation of both windows with stimuli that were equal in amplitude but varied in phase between 0° and 180° resulted in a ‘vector’ summation of the cochlear potentials in such a way that for 0° phase difference the summation was zero and for 180° phase difference the summation resulted in a 6 dB higher cochlear potential than for each of the stimuli alone on one of the windows.
This experiment repeated by Voss et.al. in 1996 confirmed the findings of Wever and Lawrence.
Moreover they found that both the DC and the AC component of the cochlear microphonics showed an identical behavior. Except for the ratio between the DC and the AC potential amplitudes, which showed an approximately 40 dB difference in favor of the DC component.
The conclusion that can be drawn from these experiments is that a resulting two times higher stimulus in perilymph movement – evoked by simultaneously stimulating both oval and round windows with sound vibrations in amplitude equal but with a 180° phase difference – create a 4 times higher change in cochlear microphonics. Due to the fact that the reported 6 dB equals a factor of 4.
Which means that the changes in cochlear microphonics are proportional to the square of the perilymph velocity in scala tympani and scala vestibuli.
The final conclusion of these experiments, according to the view of Heerens & de Ru is that the incoming sound pressure stimulus is differentiated – according the transfer from sound pressure to perilymph velocity – and squared – according the transfer from perilymph velocity to cochlear microphonics.
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