The cochlea has a complex physical structure, with multiple membranes and fluid-filled chambers.
Therefore to explain the separation of frequencies along the basilar membrane of the cochlea is complex to. Sure, there are a lot of very general descriptions (even the answer of theblackcat) and a lot never go into the actual physics of the system.
This second answer of my, again very long, covers in every detail the actual physics of the system.
More than 50 years after Georg von Békésy became a Nobel price laureate we still use a two compartment model, for our three compartment hearing sense:
The scala media is expected not to play a significant role and is therefore in models reduced to a 'thin baslilar membrane' (even in the answer of theblackcat).
And it is hypothesized that a traveling wave, running from base to apex and counting several wavelengths in its pathway in the cochlea to a place on the basilar membrane where resonance fits the best, is supposed to transport the sound energy:
Now most of his (Georg von Békésy) hypothese appeared to be incorrect. His traveling pressure wave is modified as well.
It was Georg von Békésy, who first observed and described a kind of „wave propagation‟ over the basilar membrane when the cochlea was stimulated with an acoustic vibration. A „wave‟ that always runs from base – the round window – to apex – the helicotrema.
He also noticed that the properties of the basilar membrane were quite remarkable. Smaller in size and mechanically more rigid near the base, broader and more flexible towards the apex.
A logarithmic, frequency-location relation was observed along the basilar membrane known as the tonotopic frequency distribution. This observation has led to general agreement among cochlear experts, supported by an overwhelming amount of experimental evidence, that higher frequencies are detected near the base, whereas the lower frequencies evoke the most mechanical activity near the apex.
As a result of his experiments with cochleae which he stimulated with extreme intensity, von Békésy formulated the hypothesis that the observed „propagating wave‟ – which evidently moves faster near the base with a smaller wavelength, and thereafter gradually decreases its propagation velocity and increases in wavelength towards the apex, finally even stops there, (at the place) where detection of the characteristic resonance frequency is found – carries acoustic energy to the place of detection.
Von Békésy also executed experiments with a number of mechanical models. In these models he replaced the scala media, which is confined by the extremely thin Reissner membrane on one side and the much more rigid basilar membrane on the other, by means of a single flexible membrane dominated by the properties of the basilar membrane.
In order to justify this concept he formulated the hypothesis that the Reissner membrane must be so thin and flexible that it cannot possibly influence the hydrodynamic behaviour of the perilymph within the cochlea.
Even though this concept has been adapted by a few alterations and hypothesis that are better adapted to current thinking, the basic ideas of wave propagation and the replacement of the scala media by just one single, flexible membrane remain to be cornerstones for the hypothetical explanation of signal transfer within the cochlea.
Several attempts have been made by cochlear and mathematical experts to develop theoretical models that combine the acoustic stimulation of the ear by means of a travelling wave, (carrying acoustic energy) with the detection of signals in the organ of Corti.
All these attempts failed to lead to an ultimate solution that is free of anomalies.
This failure can be explained by one important reason.
Namely, both of these assumptions:
The existence of the Reissner membrane can be neglected while the hydrodynamic behaviour is both observed and explained, and
The propagating wave must, somehow, carry the frequency dependent sound energy to the corresponding location of resonance on the basilar membrane,
are at variance with the fundamental rules of physics.
A correct application of physics leads to a totally different working model of our hearing sense.
Taking a lot of considerations and objections:
From a physics point of view against the current hypothesis of basilar membrane stimulation, one can take a lot of considerations and objections.
With a reference to:
In the manuscript "Applying Physics Makes Auditory Sense." the authors explain and here I substantiate their statements:
The enervation of the general assumption:
The existence of the Reissner membrane can be neglected while the hydrodynamic behaviour is both observed and explained.
In literature one finds many images of the cross section of the cochlear partition (the Cochlea cross-section image in the answer of thetheblackcat is such an image) clearly demonstrating that the Reissner membrane is not a loose piece of tissue. Although thin, it can be regarded as a flat structure in a normally functioning cochlea, which is an indication that some stretching must be present in it.
In patients suffering from Ménière's disease, endolymphatic hydrops causes the Reissner membrane to balloon, clearly showing pressure on the Reissner membrane.
This implies however, that if the movement of perilymph is parallel to the surface of the Reissner membrane, this membrane is not capable of lateral movement. Not, unless local stress forces are evoked in the Reissner membrane. For that purpose, the non-viscous perilymph fluid would have to develop shear stress forces in the interface of the Reissner membrane, which is not possible.
Within the science of physics we know of no such mechanism that would allow for the transfer of this lateral movement to the endolymph in the interior of the scala media, which is on the other side of the Reissner membrane, as if the Reissner membrane simply does not exist. Therefore, the hydrodynamic behaviour on either side of the Reissner membrane will always remain to be different.
It follows, that ignoring the existence of the Reissner membrane is fundamentally incorrect.
Any results of theoretical attempts to clarify cochlear functioning, based on a cochlear model that consists of two channels separated by a thin, flexible partition, ignoring the scala media as a third channel filled with endolymph in between both perilymph ducts, are therefore, also fundamentally incorrect.
The enervation of the general assumption:
The travelling wave must, somehow, carry the frequency dependent sound energy to the corresponding place of resonance on the basilar membrane,
Starting with the observations of Von Békésy, the „travelling wave‟ phenomenon has always been ascribed to the effects of sound energy transportation.
Even though from time to time, serious doubts with regard to this hypothesis have arisen – even Wever, Lawrence and Von Békésy himself – and other critical researchers have voiced the opinion that perhaps the „wavy character‟ of the basilar membrane must be found in a different cause, not one of them has ever developed an analytical calculation model accounting for all the remarkable properties of the basilar membrane. Cochlear scientists have always shared the expectation that developing such a model would be extremely complicated, and would therefore lead to unreliable results. [It is known that Von Békésy would disparagingly dispose of any such attempts as „armchair theories‟ that would not lead to anything of use.]
In a 1954 paper, Wever, Lawrence, and von Békésy reconciled some of their views on the nature of the traveling wave. They stated that when the cochlea is stimulated with a tone, a BM "displacement wave seems to be moving up the cochlea. Actually...each element of the membrane is executing sinusoidal vibrations...different elements...executing these vibrations in different phases. This action can be referred to as that of a traveling wave, provided that...nothing is implied about the underlying causes. It is in this sense that Békésy used the term ‘traveling wave’..." [pp. 511-513 of Wever et al. (1954)].
Unhindered by his disdain; as always following the curiosity that leads the way in science:
One can do the following math:
Start by calculating the sinusoidal pressure stimulation with frequency, which uniformly acts on the basilar membrane, while this membrane is infinitesimally divided into an array
of individual resonators with a logarithmically decreasing resonance frequency from base to apex.
The reason for this uniform pressure stimulation is found in the fact that it has shown that the perilymph moves as a whole fluid column along the front side of the basilar membrane, thus resulting in uniform pressure effects on the basilar membrane as well.
Making use of complex function theory and conformal transformations this general vibrational transfer model of the basilar membrane, despite its complexity, offers an analytical solution.
What's more: this solution has led to a very useful result:
And it is in accordence with what Ren and his team observed with their direct laser interferometer measurements of basilar membrane movements.
Ren’s unintentional attack on Von Békésy’s “Traveling Wave Theory”
The paper of Ren is:
Longitudinal pattern of basilar membrane vibration in the sensitive cochlea
Proceedings of the National Academy of Sciences - pnas.org
PNAS | December 24, 2002 | vol. 99 | no. 26 | 17101-17106.
Experiment: Laser interferometrical measurements of the basilar membrane movement.
In the 13,3 – 19 kHz area of the basilar membrane of a gerbil.
Results: The movement of the basilar membrane, from the higher frequency side towards the lower side, is restricted to 300 μm on both sides of the point of maximum activity. The shape of the movement was exactly symmetrical around this point.
The authors of the manuscript "Applying Physics Makes Auditory Sense." have actually paid rather a lot of attention to the form of displacement, which corresponds with the form that Ren et al. have actually measured.
So, there is a discrepancy between the assumed travelling wave from current theories and the experimental results by Ren et al.
In their experiments Ren et al. they observed a short „wave pattern‟, symmetrically divided on either side of the point of resonance. What's more, according to Ren et al, the movement of this observed wave pattern along the basilar membrane, running from base to apex, did not decrease in speed.
According the manuscript "Applying Physics Makes Auditory Sense.":
Due to the peculiar basilar membrane resonance possibilities found in practice, a uniform sinusoidal pressure stimulus results in a mirror symmetrical phase wave pattern that shows a propagating wave running from base to apex. And this waveform on the basilar membrane is identical to that which Ren et al. observed in their laser interferometer experiments on gerbils.
The reason for this phase dependent behavior is explained in more general terms in the manuscript "Applying Physics Makes Auditory Sense.".
A detailed mathematical explanation and analytical calculation has been excluded from that manuscript – but is available.
The insight that the pressure stimulus on the basilar membrane is regarded to be uniform is part of the next:
The Bernoulli effect:
Bernoulli effect is mostly known under steady flow conditions.
But that do not exist within the cochlea.
But the Bernoulli effect is not only known under steady flow conditions.
And it is not so that the Bernoulli effect can only be valid under steady flow conditions.
In the cochlea:
Therefore, the quasi-static approach, a quasi-stationary, in the concept: quasi-static relation,
the quasi-static solution, the quasi-static Bernoulli, in quasi-static form.
It is non stationary within the cochlea.
Let me clarify (like the authors do in that manuscript) why 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 of the cochlea we do not have to make use of Navier-Stokes equations at all.
The basilar membrane moves in reaction to the sound energy signal, which is generated by the quasi-static Bernoulli effect.
According to Bernoulli's law, this pressure difference on either side of both the Reissner membrane and basilar membrane is represented by:
pressure difference = - 1/2 (the density in kg/m³ of the perilymph) times (the velocity of the perilymph in m/s)^2
An erroneous assumption and point of departure – encouraged by the advise of cochlear experts – is the commonly accepted two compartment concept, in essence, ignoring the existence of the endolymph filled scale media between the two with perilymph filled channels, the scala vestibuli and the scala tympani. The latter, unavoidably lead to an incorrect hydrodynamic concept.
The fact is then that the “physics” in these current theories are clearly at variance with the general laws of physics:
Ignoring the hydrodynamic role of the Reissner membrane in the cochlear mechanics is erroneous according to general physics.
Overlooking the fact that the cochlear potential changes only when the push-pull of the perilymph velocity is generated by a sound signal, which means that the incoming stimulus is differentiated, constitutes an omission in terms of general physics.
Not taking into account that the increase of these cochlear potentials with 6 dB in case of a doubled incoming stimulus signal, signifies that there is a quadratic relation between the sound stimulus and the cochlear potentials, is another such omission.
- The common assumption that acoustic frequencies evoke waves in the perilymph channel, while the length of this channel is only a fraction of the wavelengths that can be generated, is erroneous. The existence of these waves is at variance with the general laws of physics.
regarding the Bernoulli effect:
All conditions in the cochlear channel are such that the quasi-static solution, which is equal to the use of the Bernoulli equation, applies perfectly.
The authors make use of the – from literature, such as Von Békésy – known mechanical properties of the basilar membrane, which lead to a logarithmic distribution of the resonance frequencies over the basilar membrane. Nothing more. If subsequently, the deflection profile of the basilar membrane is calculated, the absolute deflection remains dependent on the size of the stimulus, and therefore remains arbitrary.
In the manuscript "Applying Physics Makes Auditory Sense." the authors clearly describe that the basilar membrane and the Reissner membrane both move outwards from the scala media and thus away from each other.
Indirectly (outside their manuscript) they argue the fact that apparently the scala media must expand, however that the incompressibility of the endolymph prevents this expansion. Which, by the way, would have been correct if there would not be a connection between the scala media and the saccus endolyphaticus within the cerebrospinal cavity.
In their model of the auditory sense:
The saccus endolymphaticus – situated within the cerebrospinal cavity, an environment that is not subject to pressure variations such as the scala media is, but is still capable of adapting itself – functions as an expansion vessel and maintains the supplementation of the endolymph in the somewhat expanding scala media.
The fact is then that the scala media is capable of reacting to pressure variations, even though it's content is incompressible
Yes indeed, their proposed theory is completely different from the current theory. This is why, in an attempt to explain, they provided a new analysis. From this analysis that has been held against the light with regard to general physics, it becomes crystal clear that current theories are based upon interpretations of oversimplified and therefore erroneous working models, in turn leading to hypothesis and formulations that are at variance with general physics.
When one finds that an overwhelming part of literature departs from a model of two compartments, while it is clearly evident that it concerns a three compartments concept, this inevitably leads to an erroneous hypothesis. If then, finally, after much verification by erroneous interpretations of experimental results, one declares the theory departing from two compartments valid, one still cannot speak of a „large body of evidence‟.
Therefore: radically new ideas refreshing, I would like to emphasize that the authors have covered: all of these items. To mention just a few:
The claim that the auditory sense differentiates and squares and that we therefore receive the sound energy frequency signal in the organ of Corti, is completely based on the results published by Wever and Lawrence in 1950.
The „travelling wave‟ along the basilar membrane can be described as a passive reaction of a complex organised second order resonance system, and is therefore a phase wave in the cochlea, that will always run from base to apex, completely in accordance with the referred experimental observations by Ren et al.
An explanation that can satisfy the anomaly presented by the absence of the backwards travelling waves in the DPOAE experiments. Therefore, this anomaly that was acknowledged by cochlear experts like de Boer, still remains unsolved. It stands to reason; as sound energy transporting travelling waves simply cannot exist in the cochlea.
Whether the Reissner membrane is flexible or rigid in fact does not contribute much to its hydrodynamic behaviour. If the Reissner membrane were infinitely flexible –therefore deformable in the membrane surface itself - it would not exert a dominating influence on the hydrodynamic behaviour of either the perilymph moving along the membrane in the scala vestibuli nor on the immobile endolymph on the other side in the scala media. Precisely because of this lack of extreme deformability in the membrane surface the scala vestibuli and the scala media may not be considered as one hydrodynamic unit. To understand this better: one can perform physics at the kitchen table.
So, basic physics which can be performed at the kitchen table. You only have to dip a
wire ring into a soap solution, create a soap film, and then softly blow air towards the surface of the soap film. You will see the soap film bending away from you, which is experienced as completely normal behaviour.
When you gently continue to blow air sideways along the soap film you will notice the extreme deformability of the surface of this soap film. The liquid in the soap film starts to move very easily, parallel to the direction of the airflow. This causes the liquid in the soap film to be pushed in an outward direction, while the soap film rapidly decreases in thickness and subsequently looses its structure and bursts.
When however, you blow in short puffs on one side of that soapy film, you at once realize that:
The soap film, which is many times more deformable and flexible than the Reissner membrane – and should therefore be better suited to the hypothesized behaviour that Von Békésy describes – each time bulges in the direction of that side where the pulsating air flow passes by, and will then be much more resistant to bursting.
This bulging of the soap film towards that side of the soap film where the airflow passes by is exactly what the quasi-static Bernoulli effect predicts.
Based on physics' principles, when correctly applied, it is entirely impossible that either a slow wave or even a fast wave exists in the cochlea.
I can wonder, when did calculating an erroneous model ever lead to correct results?
The reproache for the wavy behaviour within the cochlea,
to introduce a three compartment model into the research.
In essence, new calculations are based on a three-compartment model, for which the authors assume as point of departure that the changing and location dependent pressure difference over the basilar membrane, between the scala tympani and the scala vestibuli, must be calculated on a three-compartment model.
Previously, it has been discussed and shown that the Bernoulli effect may be applied for all frequencies within the cochlea, be it in quasi-static form.
So one should not depart from the erroneous two compartment model, whereas in view it most definitely should be a three compartment model.
The basilar membrane moves in reaction to the sound energy signal, which is generated by the quasi-static Bernoulli effect.
How do we have to interpret that “wavy” movement of the basilar membrane?
In this we have to observe the following facts in physics:
In a medium [ gas, liquid, solid material ] there exists a uniform relation between the propagation velocity v of sound or vibration, the frequency f and the wavelength λ of the sound or vibration wave:
v = f × λ
v is lowest in gasses: In air 330 m/s
v in water but also in perilymph 1500 m/s
v is highest in solid material to ca. 8000 m/s
Together with the lowest [ 20 Hz ] and highest [ 20.000 Hz ] sound frequencies that we are able to hear, the wavelength varies in the perilymph from 75 meter to 7.5 cm
Always significantly larger than the size of the cochlea.
In the much shorter perilymph duct there cannot run a “sound wave”.
The perilymph between oval and round windows is just able to move forwards and backwards as a whole.
Tissue around the perilymph channel behaves more like a solid material than like a liquid.
That tissue needs a larger size for a traveling wave.
There cannot propagate a traveling wave inside the cochlea.
But what kind of movement is observed then ?
Therefore we must observe at first the way of movement of a singular resonator.
A resonator exist of a body connected to a spring, and is possessing in practice also damping.
If the body is given a deflection in opposite direction to the spring influence and that body is released, it will move harmonically with descending amplitude around the equilibrium point.
The frequency in that case is known as resonance frequency fr
Let us observe the reaction of a spring-mass-system
on a periodic stimilus
If the resonator is brought into a vibrating movement, then three different situations can exist, dependent on the relationship between stimulus frequency f and resonance frequency fr :
f < fr : reduced in phase movement, with phase angle: 0
f = fr : increase due to resonance but also a phase retardation with phase angle: ½ π
f > fr : strongly reduced movement in opposite direction with phase angle: π
Followed by the remarkable mechanical setup of the basilar membrane:
This basilar membrane [ BM ] exists of an array of small resonators, that have gradually decreasing resonance frequencies from the round window up to the helicotrema.
And then in case of an everywhere equal in phase stimulus on the entire BM, the following is happening:
All parts of the BM having fr > f : move in phase with the stimulus.
That movement becomes larger if fr approaches f closer and will retard gradually in phase.
In case of resonance a large movement is and there exist a phase retardation of ½ π
All parts of the BM with fr < f are more and more moving in opposite phase with the stimulus and with a growing decreasing in deflection.
And what phenomenon is comparable to this?
The “wave” in the stadium!
And dependent on the quality factor in resonance, strongly coupled to the rate of damping, the moving area becomes smaller, while the maximum deflection becomes larger.
On theoretical grounds it is no mystery that this “wavy movement” of the BM is always running from the round window [base] towards the helicotrema [apex] of the cochlea.
It is a locally bound reaction behavior on a universally existing stimulus.
Using the material specifications this behavior can be calculated in a perfect way.
If we calculate the phase relations of that same second order resonance system with the equation of phase, we find that for membrane resonance frequencies higher than the stimulus frequency, the phase of the membrane movements equals the phase of the stimulus frequency.
For membrane resonance frequencies that are lower than that of the stimulus frequency, the movements of the basilar membrane show a retarded phase shift of 180°.
The phase for the basilar membrane movement at center frequency is retarded over 90°.
This means that the auditory nerve receives the final signal, almost exclusively, from the contributions in the center frequency region. The contributions of the two flanks however, cancel each other due to their identical amplitude and opposite phase.
This mathematical calculation shows for the logarithmically distributed local resonance frequencies fc of the basilar membrane, the response characteristic that Ren observed in his experiments on gerbils: a very restricted symmetrical local movement phenomenon, which travels along the basilar membrane.
In this opinion this phenomenon is erroneously interpreted as evidence for a ‘traveling wave’ along the basilar membrane. That it is not a traveling wave, but a ‘phase wave’, that consists of coherent place dependent phase shifted local reactions to a stimulus that is simultaneously present throughout the basilar membrane.
Which results in the following animation: