Applying physics makes auditory sense

Publication

1. Willem C. Heerens & Jacob Alexander De Ru, Applying physics makes auditory sense : a new paradigm in hearing, 2010

Applying physics makes auditory sense : a new paradigm in hearing, 2010 ↓ Download full-text PDF  

Summary, 2010 ↓ Download summary PDF  

Toepassen van Fysica Zinvol bij het Horen : Een Nieuw Gehoorparadigma, 2010 ↓ Download volledige-tekst (Dutch) PDF  

Appendices

I. Heerens W, Mangelinckx Y, de Ru J. Verification of calculations of residual pitch and beat phenomena by the reader, 2010 ↓ Download A I PDF  

II. Heerens W, Mangelinckx Y, de Ru J. The residual pitch and beat phenomena that can be heard in practice by the reader, 2010 ↓ Download A II PDF  

Heerens W, Mangelinckx Y, de Ru J. Perception calculations, 2010
This associated software: "Perception calculations" together provide you with a tool to personally verify the predicted residual pitch and beat phenomena described in chapter 3 of the booklet.
This program is ONLY available for computer systems running under Windows. ↓ Download associated software  

We also present the composed sound fragments without this program, so in case you are not able to use the program, you are invited to download the composed sound fragments to personally verify the predicted residual pitch and beat phenomena.

Presentation of composed sound fragments. E.2. Pitch perception in incomplete harmonic sound complexes.
↓ Download E2examples  
Presentation of composed sound fragments. E.3. Residual pitch perception in unresolved enharmonic tone series.
↓ Download E3examples  
Presentation of composed sound fragments. E.4. Addition of harmonics and their influence on beat phenomena.
↓ Download E4examples  
Presentation of composed sound fragments. E.5. Modifying a beat frequency by adding a low frequency stimulus.
↓ Download E5examples  
Presentation of composed sound fragments. E.7. Is pitch shift in an enharmonic tone complex an illusion?
↓ Download E7examples  
Presentation of composed sound fragments. E.8. Infrasound can also be heard according to the squaring principle.
↓ Download E8examples  

---

I. Heerens W, Mangelinckx Y, de Ru J. Verification of calculations of residual pitch and beat phenomena by the reader, 2010

II. Heerens W, Mangelinckx Y, de Ru J. The residual pitch and beat phenomena that can be heard in practice by the reader, 2010

---

---

Daarom zijn wij er van overtuigd dat ons cochleaire model voorziet in een duidelijke oplossing voor de nog altijd bestaande problemen die ook genoemd worden in Alain de Cheveigné’s historische overzicht in zijn publicatie getiteld: "Pitch percep-tiemodellen van het begin tot heden", welke gepubliceerd is op Internet [51]. Omdat de conclusies van De Cheveigné in deze publicatie van 2004 zo nauw in verband staan met hetgeen wij gevonden hebben, kunnen wij niet nalaten om het volgende daaruit te citeren:

• Therefore, we are convinced that our cochlear model provides a clear solution to the still existing problems that were also mentioned in Alain de Cheveigné’s historical overview entitled: "Pitch perception models from origins to today" which is published on the Internet [51]. As De Cheveigné’s conclusions in this paper of 2004 are so closely correlated to our findings, we cannot possibly withhold the following citation:
  

• --- Modern ideas reincarnate older ideas, and their roots extend
  as far back as records are available. Models that are in competition
  today may have common roots.
  The historical approach allows commonalities and differences to
  be put in perspective. Hopefully this should help to defuse sterile
  controversy that is sometimes harmful to the progress of ideas. It
  also may be of use to newcomers to the field to understand, say,
  why psycho-acousticians insist on studying musical pitch with unresolved
  stimuli (that sound rather unmusical), why they add low
  pass noise (which makes tasks even more difficult), etc.
  The good reasons for these customs are easier to understand with
  a vision of the debates from which present-day pitch theory
  evolved. ---

--- Moderne ideeën zijn reïncarnaties van oudere ideeën en hun wortels reiken zover als er historische documenten daarover beschikbaar zijn. Modellen die heden ten dage met elkaar wedijveren, kunnen best gemeenschappelijke wortels hebben. De historische benadering maakt het mogelijk overeenkomsten en verschillen in het juiste perspectief te zetten. Hopelijk kan dit helpen om onvruchtbare tegenstellingen te elimineren die soms schadelijk zijn voor de ontwikkeling van ideeën. Het zou ook van nut kunnen zijn om laten we zeggen voor nieuwkomers te begrijpen, waarom psychoakoestici erin volharden om muzikale toonhoogte te bestuderen met onopgeloste stimuli (die nogal onmuzikaal klinken), waarom zij laagfrequente ruis toevoegen (wat de taken zelfs nog ingewikkelder maakt), enz. De goede redenen voor deze gebruiken zijn gemakkelijker te begrijpen indien wij de debatten in ogenschouw nemen waaruit de hedendaagse toonhoogtetheorie zich ontwikkeld heeft. ---

--- Moderne ideeën zijn reïncarnaties van oudere ideeën en hun oorsprong reikt zover als historische gegevens daarover beschikbaar zijn. Modellen die nu onderling tegenstrijdig zijn, kunnen best gemeenschappelijke wortels hebben. De historische benadering maakt het mogelijk onderlinge overeenkomsten en verschillen in het juiste perspectief te zetten. Hopelijk kan dit helpen om onvruchtbare tegenstellingen te elimineren die soms schadelijk zijn voor de ontwikkeling van ideeën. Het kan ook van nut zijn voor nieuwkomers in het vakgebied zogezegd (wat de taken zelfs nog ingewikkelder maakt) , enz... De goede redenen voor deze gebruiken zijn gemakkelijker te begrijpen met een blik op de debatten waaruit de huidige pitch theorie is ontwikkeld. ---

---

So, also: "... - that is how pitch is perceived." ?

with a vision of the debates from (E) to (F) in Figure 1 Spectral approach from De Cheveigné A. (2005)

If I also look at that one complicated pitch perception example, namely, the complicated pitch perception example (E) given by De Cheveigné

De Cheveigné A. (2005) Pitch Perception Models. In: Plack CJ, Oxenham AJ, Fay RR, Popper AN, editors. Pitch: Neural Coding and Perception: 169 - 233. New York: Springer Science + Business Media, Inc. ISBN 10: 0-387-2347-1.

with a vision of our debates from W5 to W7 the corresponding and resulting sound energy frequency spectrum (W7), according to our paradigm, can be calculated:

For the complicated pitch perception example given by De Cheveigné [40] the corresponding and resulting sound energy frequency spectrum can be calculated.This is shown W7.
Again, because our auditory sense appreciates the 1/f criterion for tone contributions in the sound pressure tone complex, this is chosen as a constraint in W5.

---

• A New Explanation of Pitch Perception Phenomena described in Literature. ↓ Download A New Explanation of Pitch Perception Phenomena described in Literature PDF  



• Remarkable Pitch Perception Phenomena ↓ Download Remarkable Pitch Perception Phenomena PDF  

---

Video All this results in the explanation of a number of important auditory phenomena.

( If you don’t see the video image below: refresh the page or clear your cache. )(video 5)

---

Please, do the following series experiments:

And let me inform you in advance that the remarkable results heard by me (Heerens) were also heard without exception by all other not specifically trained observers I had asked to do these experiments as well.

With my (Heerens) age plus a severe loss of hearing due to Meniere’s disease since 1985 I can still hear a 10 kHz pure sinus tone beep.

However if I take for instance the frequency series of:

            10000+10004+10008+10012+10016+10020+10024 Hz

with all sine functions I clearly observe a 4 Hz beat in that beep tone.

And then with alternating sine cosine contributions. I clearly observe an 8 Hz beat in the same beep tone.

This phenomenon – known in literature for higher difference frequencies like 100 Hz between successive lower frequent harmonics – cannot be explained by simply adding up in a linear way all the contributions to a total of one single frequency of  10012 Hz, that is modulated in such a simple way.

This because such an addition results in the combination:

And if we look at the modulation factor:

we see here three modulation frequencies:  4, 8 and 12 Hz , each with amplitude 2 as equal weight factor.

Together with the constant contribution with weight factor 1 this can only result in a weird mixing in modulations
of 8 + 16 + 24 Hz.

[Here a doubled frequency contribution to the beat exists for each contribution, because the different modulation factors reaches between + and – 100%]

This calculated sum signal is definitely not what we hear.

Another confusing fact is found if we observe the way these seven frequency contributions are related to the beat of 4 Hz we hear in case of all sine function contributions, and the beat of 8 Hz we hear in case of alternating sine / cosine contributions to the sound complex.

From Internet I found the figure out of the presentation of A. Foulkner about the perception of pitch :

http://www.phon.ucl.ac.uk/courses/spsci/audper/Pitch_AUDL4007_2010.pdf

There in slide 25 a clear picture of resolved and unresolved frequencies is given:

The 10000 and following frequencies can be seen as the 2000^th  to 2006^th harmonics of the ‘pitch’ of 5 Hz.

So no question about it: The entire sound complex is completely unresolved.

Changing the all sine complex into the alternating sine / cosine complex would result for linear summation to:

An even much stranger modulation of a 10012 Hz beep, including a phase shift, that can never create the 8 Hz beat we actually hear.

In other words: It is absolutely clear that there happens something different inside the cochlea.

I say in the cochlea, because we cannot expect that our brain simply calculates such phenomena out of the extremely poorly to the frequencies correlated firing rates in the nerve fibers. Firing rates that have no frequency relation to the offered sound and appears more like stochastic contributions.

And if we compose the following sound complex with all sine functions:

            10000+10004.0625+ 10008+10012.0625+ 10016+10020.0625+10024  Hz

[ So each of the 2^nd , 4^th and 6^th contributions are shifted by an extreme small amount of 1/16^th of one Hz. ]

we hear a very peculiar sound:

A beep with a variable beat, that alternates every 8 seconds  from a 4 Hz beat to an 8 Hz beat.

Now changing the sine functions of the 2^nd , 4^th and 6^th contributions into cosine changes not really the alternating beat in the then heard sound.

What we hear in both cases is a sound fragment with a period of 8 seconds that alters from a 4 Hz beat into an 8 Hz beat.

The changing of the ‘mistuned contribution’ from f. i. 10012.0625 into 10011.9375 Hz – or one of the other two contributions, only changes partially the depth of the beat, but not the rhythm of it.

And when all these three contributions are ‘mistuned’ to lower frequencies:

            10003.9375 Hz + 10011.9375 Hz + 10019.9375 Hz

the modulation exactly sounds like that with the ‘mistuned’ frequencies:

            10004.0625 Hz + 10012.0625 Hz + 10020.0625 Hz

 

And finally the last step:  We can even raise all frequency components with an amount  , not equally to the difference frequency of 4 Hz, what makes that the series aren’t even extreme high harmonics of the 4 Hz difference frequency.

And even this doesn’t change any noticeable thing in the heard sound.

Let me state again:

A sound complex, in this example existing of seven completely unresolved contributions evokes hearing sensations that are held for impossible in the current pitch perception theory.

In the current  hearing theory, in both all sine as well as alternative sine – cosine contributions, you wouldn’t expect a double beat phenomenon.

To my knowledge this is a brand new anomaly within the paradigm of the current hearing theory.

But these phenomena are completely calculable and predictable as well in all details if we apply the hearing paradigm that I have formulated in the booklet, mentioned earlier:

Applying Physics Makes Auditory Sense

I invite you to verify or if you wish to falsify these experimental results by carrying out the experiments described by me.

I hope this will cause a lot of astonishment and excitement.

 

Kind greetings

 

Pim Heerens

---

Experiments Perception calculations

Verification of calculations
of beat phenomena

by Willem Christiaan Heerens

One more experiment:

You can carry out the following series of experiments:

Please download the software program with which these sound complexes can be properly
calculated in the form of wav files from here:

↓ Download associated software  

This program is ONLY available for computer systems running under Windows.

[ NOTE: The standard setting in the 1/f mode in this software program
takes care that all the individually primary calculated frequencies
contribute equal energy to the resulting sound pressure signal.
This condition is very important for the influences on pitch calculations in
case higher values of the differences between contributing frequencies exist. ]

Entirely based on the premises of the new paradigm,
Pim Heerens has calculated a number of predictable sound phenomena
by using the following frequencies together with prescribed phase relations
in a standard summation procedure to compose a Fourier series:

1:

10000 + 10004 + 10008 + 10012 + 10016 + 10020 + 10024 Hz
Where all the contributions are sine functions.

Our paradigm predicts: an undisputable beat of 4 Hz in a high beep tone.

2:

10000 + 10004 + 10008 + 10012 + 10016 + 10020 + 10024 Hz
Where the contributions are successively alternating sine and cosine functions.

Our paradigm now predicts: an undisputable beat of 8 Hz in the same high beep tone.

3:

10000 + 10004.0625 + 10008 + 10012.0625 + 10016 + 10020.0625 + 10024 Hz
Where all the contributions are sine functions.

Our paradigm now predicts: a beep, in which an undisputable beat exists
that changes every 8 seconds from clearly 4 Hz to 8 Hz and then reverses
again to 4 Hz. So the beat pattern has a period of 8 seconds caused by the
systematic mistuning of 1/16 = 0.0625 Hz.

Additional changes in the mistuning, like for instance from 10004.0625 into
10003.9375 Hz, of either one, two or three of the mistuned frequencies are
predicted to give the same results in the beat pattern as experiment 3.

---

If you carry out the same series of experiments with a start frequency of
1000 Hz instead of 10000 Hz, you will hear the same series of beat
phenomena, but now with the lower beep of the 1012 Hz instead of the 10012 Hz beep.
Even if you go down with the start frequency to 200 Hz or 400 Hz you will
still hear the same beat phenomena, but now with the low humming tone of
200 Hz respectively with the one octave higher humming tone of 400 Hz.

Hence it is a perception phenomenon that appears all over the entire
auditory frequency range.

Once these beat phenomena are verified as really existing for every
listener with a reasonable normal hearing, for the current paradigm
this is a very serious anomaly.

What for instance is observed from the following combinations of frequencies:

10000 + 10004.0625 + 10008 + 10012.0625 + 10016 + 10020.0625 + 10024 Hz

2000 + 2004.0625 + 2008 + 2012.0625 + 2016 + 2020.0625 + 2024 Hz

400 + 404.0625 + 408 + 412.0625 + 416 + 420.0625 + 424 Hz

having all sine or all cosine contributions, is that they will have an
average frequency – 10012; 2012 respectively 412 Hz – with a beat rhythm
of:

4 – 8 – 4 – 8 – 4 – 8 – 4 – 8 – 4 Hz within 32 seconds, so a period of 8 seconds.

While having alternating sine – cosine – sine ... or cosine – sine – cosine ...
contributions, they get a beat with the opposite sequence in the rhythm
of:

8 – 4 – 8 – 4 – 8 – 4 – 8 – 4 – 8 Hz within 32 seconds, so with a period of 8 seconds.

• In these series you can of course try to attribute the beat phenomena
  to ‘special combinations’ of these frequency contributions to eliminate
  the problems that arise when the traveling wave model is applied.
  
  Well in order to enervate these suggestions in advance, let us add another
  experiment with which this (unfair attributing to) is absolutely impossible.

---

Let us chose the following frequency contributions:

Once these beat phenomena are verified as really existing for every
listener with a reasonable normal hearing, for the current paradigm
this is absolutely a very serious anomaly.

A series of five tones existing of:

8009 + 8011 + 8013 + 8015 + 8017 Hz
with a minimal difference frequency of 2 Hz.

Where the first, second and fifth contributions are prime numbers, the
8013 Hz contribution is the product of two prime number: 3 x 2671, and the
8015 Hz contribution is the product of three prime numbers: 5 x 7 x 229.
These integer frequency contributions only have 1 Hz as fundamental.

And a series of five tones existing of:

7499 + 7501.0625 + 7503.125 + 7505.1875 + 7507.25 Hz
with a difference frequency of 2.0625 Hz. Again no fundamental.

If all those contributions have sine or cosine functions the beat
phenomenon is given by: a high beep tone with a dominant beat rhythm of:
2 – 4 – 2 – 4 – 2 Hz within 32 seconds, so with a period of 16 seconds
that is mixed with a weaker 6 Hz beat rhythm.

If you modify each of the two frequency series in sine–cosine–sine–cosine–sine
or cosine–sine–cosine–sine–cosine contributions, you will hear a beat
phenomenon in the same high beep tone, but now with a beat rhythm of
4 – 8 – 4 – 8 – 4 – 8 – 4 – 8 – 4 Hz within 32 seconds, so with a period of 8 seconds.
Now not only the 2 Hz beat, but also the 6 Hz beat is disappeared.

Now each series apart produce a pure beep tone with a 2 Hz, respectively a
2.0625 Hz beat, in case of all sine or cosine contributions and a 4 Hz
beat respectively a 4.125 Hz beat in case of alternating sine–cosine–sine–cosine–sine
or cosine–sine–cosine–sine–cosine contributions.

If we calculate the sound energy frequency spectrum, we can observe that
only the series of difference frequencies 2.0625 + 4.125 + 6.1875 + 8.25 Hz,
respectively 2 + 4 + 6 + 8 Hz, in the pair by pair combined situation
can generate the beat phenomena we can hear.

For the 10 kHz experiments the 0.0625 Hz detuning means that there exists
an accuracy in the periodicity pattern of 6.25 parts per million.

These salient auditory perception results for each chosen average frequency
over the entire auditory frequency domain are heard.

---

Experiments Perception calculations

Verification of calculations
of beat phenomena

by Willem Christiaan Heerens

One more very important experiment:

 

At high frequencies, do we perceive differences between random and deterministic components?

There is a very simple answer to the question.

That answer is:

We definitely hear great differences. They depend on the ‘composition’ of the contributing sinusoids.
But also on the length of the period of listening.

And in such compositions both the choices of frequencies and phases have strong influence.

For example:

Please download the software program with which these sound complexes can be properly
calculated in the form of wav files from here:

↓ Download associated software  

This program is ONLY available for computer systems running under Windows.

[ NOTE: The standard setting in the 1/f mode in this software program
takes care that all the individually primary calculated frequencies
contribute equal energy to the resulting sound pressure signal.
This condition is very important for the influences on pitch calculations in
case higher values of the differences between contributing frequencies exist. ]

Please calculate with high resolution the following three compositions,
using five sinusoids:

1. 10,000 / 10,002 / 10,004 / 10,006 / 10,008 Hz. All sine contributions.

In that case you will hear the high tone that corresponds with 10,004 Hz but
with a strong beat of 2 Hz.

2. 10,000 / 10,004 / 10,008 Hz. All three sine contributions.
10,002 / 10,006 Hz. Both cosine contributions. So a 90 degree phase shift.

In that case you will hear the high tone that corresponds again with 10,004 Hz
but now with a strong 4 Hz beat.

3. 10,000 / 10,002.0333 / 10,004 / 10,006.0333 / 10,008 Hz. All sine contributions.

In that case you will hear the high tone of 10,004 Hz again,
but within a period of 30 seconds

and starting with a 2 Hz beat
after 7.5 seconds the beat will gradually change into a 4 Hz beat.
After 15 seconds the beat is back again at 2 Hz.
At 22.5 seconds again at 4 Hz
and after 30 seconds the composition ends with a 2 Hz beat in the 10,004 Hz tone.

If you change the sine contributions of 10,002.0333 and 10,006.0333 Hz into cosine
the composition

starts with a beat of 4 Hz,
2 Hz at 7.5 sec,
4 Hz at 15 sec,
2 Hz at 22.5 sec
and finally 4 Hz at 30 sec.

For noise filtered by a narrow band-pass around 10 kHz it is known that we
will hear just a 10 kHz tone. Nothing more.

So on the question:

For example, do we perceive a difference between a few sinusoids around 10kHz
and a band-pass filtered noise around the same frequency?

The answer is clear: Although, according to existing perception theory, the
different frequency contributions in the composition are entirely unresolved
we can hear differences related to different phase and frequency settings.

In this context we can look at August Seebeck’s statement that he published in the year 1844:

“How else can the question as to what makes out a tone be decided but by the ear?”

It was part of his answer to the erroneous hypotheses of Ohm about pitch perception in the famous
Ohm-Seebeck dispute.

And we can add the following to it:

“After verifying the sound experiments, we are of the opinion that this - Applying Physics Makes Auditory Sense - theory is representative for the working principle of the human ear and for the cochlea.”

The above described sound experiments with indisputable results are entirely
based on the hearing theory of Heerens / J. A. de Ru in the booklet:

Applying Physics Makes Auditory Sense

Based on the concept in this booklet that our hearing sense is
differentiating and squaring the incoming sound pressure stimulus, this mechanism
evokes in front of the basilar membrane the sound energy frequency spectrum.

In that case Fourier series calculations show exactly the frequency spectrum
including the 2, 4, 6 and 8 Hz difference frequency contributions. Of which
the 2 and 4 Hz frequencies are responsible for the beat phenomena.

---

The physics makes auditory sense

Applying physics

↓ Download NonstationaryBernoulli PDF  

Heerens presented in a PDF the solution of the non-stationary Bernoulli equation, that is perfectly well valid in the case of the push-pull movements of the perilymph inside the scala tympani [ST] and scala vestibuli [SV], while the in between embedded scala media [SM], filled with endolymph at rest, has substantial – and therefore not negligible – dimensions.

According to hydrodynamic rules these dimensional conditions make that the hypothesis in which both the influence of the Reissner membrane and the content of the SM can be ignored and the cochlear duct can be considered as a folded tube with only the BM as an interface in between is definitely invalid.

Well like the well-known promoter of physics, MIT professor Walter Lewin, does in his magnificent physics courses, Heerens has built his own demonstration equipment for clearly showing what happens on the walls of a duct in which an alternating flow in core direction is evoked.

The one experimental set-up is extremely simple, but therefore also highly convincing.

As can be seen in the above figure, to mimic utmost compliance in the ‘walls’ in one of the experiments Heerens has hanged on thin wires in an open frame two sheets of paper that can move freely.

Between the two he can evoke an alternating flow parallel to the surfaces of the sheets of paper with by moving up and down a spatula.

And like it is shown in the next figure he has constructed a closed loop with a tube and a bellow, the latter centrally subdivided by a plate, with which he can create a push-pull flow in the tube, while in the upper branch of the tube locally a flexible yellow membrane is mounted in the wall, which registers what happens on the wall of the tube.

In front of the membrane a wire cross is closely mounted. Striking light from above forms a bended shadow of the wire cross on the membrane if that membrane is moving away – so inwards the tube – while during movement outwards of the membrane the shadow won’t be present because the wire cross is laid on the bending membrane.

The obtained results he found in both experiments?

The evoked motion patterns are exactly identical to what could predict out of the theory Heerens has presented. The two sheets of paper are not at all moving in outward direction. They are moving in opposite direction, so towards the core line of the alternating flow. And under a steady alternating stimulus (with constant amplitude) they both do that with a stationary deflection on which an alternating deflection is superposed with doubled frequency.

This indicates that both sheets experience the influence of an alternating and in average lower pressure evoked in the space between the two sheets.

This behavior is shown in the following multi moment presentation:

The tube experiment also shows that the membrane in the wall is always moving inwards – so towards the core line of the tube. And superposed on a constant deflection inwards the membrane also deflects periodically with double frequency related to the original stimulus frequency.

This is given in the following impression:

Without any doubt this is indicating that at least squaring of the input stimulus plays a dominating role.

[Note: To make it even more convincing for everyone, see the video registration of the tube experiment.]

Video Movement a membrane. By the Bernoulli effect.

( If you don’t see the video image below: refresh the page or clear your cache. )(video 3)

For now the only clear and firm conclusion one can draw is: The medium in the tube is moving as a whole. And therefore these experimental results, in combination with the theoretical solution of the non-stationary Bernoulli equation, are one of the reasons that the transmission line concept cannot play a role in it either.

---

The second reason for rejecting the traveling wave concept is the following: Heerens also has studied the different possibilities for ‘traveling waves’ in literature. And then especially he has looked at the conditions, parameters and geometrical dimensions under which such waves can exist.

In short (you don’t need expensive literature retrievals, because you can read a summary of the possible wave forms in Wikipedia) we can state that there are three forms to distinguish:

1. Rayleigh waves

Rayleigh waves are a type of surface acoustic waves which travel on solid materials. The typical speed of these waves is slightly less than that of so-called shear waves. And it is by a factor (dependent on the elastic constants) given by the bulk material. This speed is of the order of 2–5 km/s. For a sound signal with a 1000 Hz frequency this means that the minimal wavelength is approximately 2 meter. While the BM has a length of approximately 35 millimeter, it is impossible to make a realistic combination for application in the cochlea.

Besides that Rayleigh waves are surface waves where the thickness of the material must be relatively high related to the concerned wavelength. With a fraction of a millimeter thickness for the BM you can forget that this type of wave can play a role in the BM vibrations.

2. Love waves

In the field of elastodynamics, Love waves, named after A. E. H. Love, are described as horizontally polarized shear waves guided by an elastic layer, which is "welded" to an elastic half space (so a very thick part of bulk material) on one side while bordering a vacuum on the other side. In literature can be found that the wavelength of these waves is relatively longer than that of Rayleigh waves. And also these conditions and parameters are nowhere found in the cochlear partition.

3. Lamb waves

Lamb waves propagate in solid plates. They are elastic waves whose particle motion lies in the plane that contains the direction of wave propagation and the plate normal (the direction perpendicular to the plate). In 1917, the English mathematician Horace Lamb published his classic analysis and description of acoustic waves of this type. The wave propagation velocities of the two possible modes in Lamb waves are comparable with that of the Rayleigh wave. And therefore they also don’t provide for a possible application in the traveling wave description inside the cochlea.

In other words: we also cannot make a realistic fit with Lamb waves inside the cochlea. Of course everybody can persist in believing that until now registered auditory experimental results justify the formulated hypothesis that such types of waves can exist in the cochlea.

Then however you are forced to answer the following question:

On what underlying physics grounds is it possible that material quantities and acoustic process parameters inside the cochlea can be altered in such a way that as a result the wavelength of 1.5 meter for a 1000 Hz stimulus in bulk perilymph fluid can be altered in less than 1.5 millimeter?

As can be seen from the Rayleigh, Love and Lamb waves the circumstances and material properties cannot provide for a scaling factor better than 0.5 from bulk material sound velocity to the concerned type of wave.

Be aware that inside the cochlea a scaling factor of 0.001 or even smaller will have to be possible. This can be considered as completely impossible.

What remains is that just as Heerens stated: The described non-stationary Bernoulli effect, that provides for the sound energy stimulus everywhere in front of the BM, is driving the BM vibrations.

I have always wondered about what drives BM vibrations?

It is the everywhere present sound energy stimulus that drives the BM.

 

 

 

---

↓ Download NonstationaryBernoulli PDF  












In case of flow in a tube under the material conditions incompressible and
non-viscous and a rotation free flow condition, for stationary flow counts
the Bernoulli equation. In a horizontal orientated tube gravity doesn’t
play a role, what leads to the well known equation: the decrease of the
overall existing internal pressure is proportional to fluid velocity
squared. In the case of a non-stationary flow with all other conditions
the same as above, that overall pressure inside the tube
– and thus also on its boundaries – is proportional to the time dependent
fluid velocity squared.

While for the flow, inside the perilymph duct, not just one single but all
conditions for a potential flow and thus for the analytical solution
according to Bernoulli’s relation for non-stationary flow are fulfilled.
And it is that solution based on the sound and solid use of hydrodynamic
rules and laws that is the straight forward outcome.

And that result can be summarized in the ultimate short statement that the
changes in the internal pressure everywhere in the perilymph – that moves,
or better wiggles, on the rhythm of the sound pressure in front of the
eardrum – are proportional to the corresponding sound energy.
The change in the internal pressure is a decrease proportional to the time
derivative of the sound pressure signal squared.

Based on that result we have done the series of sound experiments that are
described in Chapter 3 of our booklet and that are explained there in
detail in the Appendices. Together with the offered downloadable
calculation program for composing those sound complexes, the inquisitive
reader can verify all our results.

In all these proposed experiments the calculation of the different
contributions to the sound energy frequency spectrum resulted per
experiment in exact predictions of the final beat rhythm.

The non-stationary potential flow according to Bernoulli in the perilymph
duct, like calculated, includes that everywhere inside this fluid
there exist the balance between the kinetic energy represented by the
expression ‘1/2 rho v^2’ – or for the total perilymph volume V
‘1/2 m v^2’ and the decrease in potential energy,
given by the expression : ‘– V delta p’.
Here rho is the density of the fluid; v the fluid velocity; delta p the
pressure difference and m the mass of the fluid column.

So also here the sound energy signal is present inside the perilymph fluid.
However not in the form of an assumed traveling wave, but as a uniform
pressure stimulus all over the volume.

And therefore all the existing Fourier frequency components in the sound
energy signal are present inside the perilymph to stimulate the basilar
membrane including their relative amplitudes and their relative, but
extremely precise, phase relations.

And it is this concept that makes it possible to calculate all the
phenomena heard in the sound experiments, even if they are as weird as the
sound perceptions in the 10 kHz examples.

---

• A New Explanation of Pitch Perception Phenomena described in Literature. ↓ Download A New Explanation of Pitch Perception Phenomena described in Literature PDF  



• Remarkable Pitch Perception Phenomena ↓ Download Remarkable Pitch Perception Phenomena PDF  

---

Nietzsche’s Ode aan de Fysica

• --- Und dazu müssen wir die besten Lerner und Entdecker alles Gesetzlichen und Notwendigen in der Welt werden: wir müssen Physiker sein, um, in jenem Sinne, Schöpfer sein zu können, — während bisher alle Wertschät-zungen und Ideale auf Unkenntnisse der Physik oder im Widerspruch mit ihr aufgebaut waren. Und darum: Hoch die Physik! Und höher noch das, was uns zu ihr zwingt, — unsre Redlichkeit! --- Friedrich Wilhelm Nietzsche 1882 ‘Die fröhliche Wissenschaft’ Buch IV ‘Sanctus Januarius’

--- En daartoe moeten wij de beste leerlingen en ontdekkers worden van al wat wettelijk en noodzakelijk is in de wereld: wij moeten fysici zijn, teneinde, in die zin scheppers te kunnen zijn, — terwijl tot op heden alle waar-deoordelen en idealen gebaseerd waren op onbekendheid met de fysica of in tegenspraak met haar. En daarom, hoog leve de fysica! En leve nog hoger datgene, wat ons tot haar verplicht, — onze oprechtheid!---
Vertaling Willem Chr. Heerens 2010

Nietzsche’s Ode to Physics

• --- Und dazu müssen wir die besten Lerner und Entdecker alles Gesetzlichen und Notwendigen in der Welt werden: wir müssen Physiker sein, um, in jenem Sinne, Schöpfer sein zu können, — während bisher alle Wertschätzungen und Ideale auf Unkenntnisse der Physik oder im Widerspruch mit ihr aufgebaut waren. Und darum: Hoch die Physik! Und höher noch das, was uns zu ihr zwingt, — unsre Redlichkeit! --- Friedrich Wilhelm Nietzsche 1882 ‘Die fröhliche Wissenschaft’ Buch IV ‘Sanctus Januarius’

--- And to that end we must become the best learners and discoverers of everything that is lawful and necessary in the world: we must become physicists in order to be able to be creators in this sense, — while hitherto all valuations and ideals have been based on ignorance of physics or were constructed so as to contradict it. Therefore: long live physics! And even more so that which compels us to turn to physics, — our honesty! --- Translated by: Walter Arnold Kaufmann 1974 ‘The Gay Science’ Vintage Books ISBN 0-394-71985-9

 

---

Applying the non-stationary Bernoulli equation.

I (Heerens) have derived the analytical solution for the non-stationary non-viscous incompressible time dependent wiggle-waggle movements directed along the core of the perilymph duct. Because in that case the reduction of the complex set of Navier-Stokes equations to the non-stationary Bernoulli equation is fully permitted.

↓ Download NonstationaryBernoulli PDF  

---

A Three Compartment Cochlear Model.

---

Organ of corti operation. Inner hair cells are the leftmost row, outer hair cells are the other three rows.

By the Bernoulli effect. Scala Tympani - Perilymph duct.

---

---

The story of our project, highlighting the broad picture

Presentation: Applying physics makes auditory sense On Prezi

Applying physics makes auditory sense

Contents

      74 pages

The content of the book is divided into nine chapters. 

• Nietzsche's Ode to Physics 04
• Prologue 05
• Introduction 10
• Objections against the traveling wave hypothesis 10
• Objections against the cochlear amplifier hypothesis 12
• Objections against the existing bone conduction signal transfer hypothesis 13
• Objections against the existing description of inner ear hydrodynamic behavior 13
• In conclusion 15
• The new hypothesis 16
• Middle ear functioning 16
• Relation between sound stimulus and electrical output in the cochlea 17
• Bone conduction 19
• The hydrodynamic behavior in the cochlea 21
• Von Békésy's traveling wave versus the Bernoulli effect 21
• Transfer of sound signals in the cochlea and the Bernoulli effect 25
• Methods and experiments for verification 29
• Stimulating the cochlea with composed frequency complexes 29
• Pitch perception in incomplete harmonic sound complexes 29
• Residual pitch perception in enharmonic tone series 31
• Addition of harmonics and their influence on beat phenomena 32
• Modifying a beat frequency by adding a low frequency stimulus 33
• Schouten's explanation for the strike note of bells must be revisited again 34
• Is pitch shift in an enharmonic tone complex with equidistant frequencies an illusion? 34
• Infrasound can be heard according to the squaring principle 36
• The role of DC signals in the organ of Corti and the cochlear amplifier 38
• The organ of Corti as a highly selective frequency analyzer 40
• Resonance phenomena in the basilar membrane 42
• Basilar membrane resonance phenomena instead of travelling waves 44
• Conclusions 47
• Acknowledgements 49
• Epilogue 50
• References 51
• Appendices 54
• Verification of calculations of residual pitch and beat phenomena 54
• Introduction 54
• How to use the software program after the installation 54
• Possible calculations this program allows you to make 56
• The successive experiments 57
• The residual pitch and beat phenomena to be heard in practice 65
• Introduction 65
• Experiments 66

 

 

---

figures

figure 1

Fig. 1. Acoustic energy flow by two traveling waves explained by Lighthill

 

 

 

 

---

figure 2

Fig. 2. Schematic drawing of the ear as an externally activated spring-mass system

 

 

---

 

 

figure 3

Fig. 3. Cross section of the cochlear channel with two and three compartments functionality

 

 

---

• Therefore, we are convinced that our cochlear model provides a clear solution to the still existing problems that were also mentioned in Alain de Cheveigné’s historical overview entitled: "Pitch perception models from origins to today" which is published on the Internet [51]. As De Cheveigné’s conclusions in this paper of 2004 are so closely correlated to our findings, we cannot possibly withhold the following citation:
  

• --- Modern ideas reincarnate older ideas, and their roots extend
  as far back as records are available. Models that are in competition
  today may have common roots.
  The historical approach allows commonalities and differences to
  be put in perspective. Hopefully this should help to defuse sterile
  controversy that is sometimes harmful to the progress of ideas. It
  also may be of use to newcomers to the field to understand, say,
  why psycho-acousticians insist on studying musical pitch with unresolved
  stimuli (that sound rather unmusical), why they add low
  pass noise (which makes tasks even more difficult), etc.
  The good reasons for these customs are easier to understand with
  a vision of the debates from which present-day pitch theory
  evolved. ---

So, also: "... - that is how pitch is perceived." ?

with a vision of the debates from (E) to (F) in Figure 1 Spectral approach from De Cheveigné A. (2005)

If I also look at that one complicated pitch perception example, namely, the complicated pitch perception example (E) given by De Cheveigné

De Cheveigné A. (2005) Pitch Perception Models. In: Plack CJ, Oxenham AJ, Fay RR, Popper AN, editors. Pitch: Neural Coding and Perception: 169 - 233. New York: Springer Science + Business Media, Inc. ISBN 10: 0-387-2347-1.

with a vision of our debates from W5 to W7 the corresponding and resulting sound energy frequency spectrum (W7), according to our paradigm, can be calculated:

For the complicated pitch perception example given by De Cheveigné [40] the corresponding and resulting sound energy frequency spectrum can be calculated.This is shown W7.
Again, because our auditory sense appreciates the 1/f criterion for tone contributions in the sound pressure tone complex, this is chosen as a constraint in W5.

This is also shown in this figure.

figure 4

Fig. 4. Transfer of sound pressure signal to sound energy frequency spectrum

 

 

• A New Explanation of Pitch Perception Phenomena described in Literature. ↓ Download A New Explanation of Pitch Perception Phenomena described in Literature PDF  



• Remarkable Pitch Perception Phenomena ↓ Download Remarkable Pitch Perception Phenomena PDF  

 

 

 

---

The relationship between frequency and wavelength. And therefore: Objections against the traveling wave hypothesis

The relationship between frequency and wavelength is covered in any physics textbook.

About the interpretation of the equation:          Interpretation of the equation in which is expressed that the wave velocity equals the frequency multiplied by the wavelength [ in common wave propagation theory in physics ].

Take for example (sound):

With the traveling wave equation: the propagation velocity of the wave [in m/s] equals the frequency [in Hz] multiplied by the wavelength [in m].

This equation must be interpreted in the following way: the speed of a (sound) wave that moves through a medium isn’t dependent on its frequency and its wavelength.

The speed (of sound) – hence also the speed with which (sound) energy is transported – is a material constant and it therefore only depends on a number of properties of that medium. And the only way to change that speed is to change the properties of the medium.

Once the speed (of sound) in a medium is determined the above mentioned equation expresses the relation between the (sound) frequency and the wavelength.

The two have an inverse relationship.

Given the frequency of the wave, the wavelength is equal to the speed (of sound) in the medium divided by the frequency.

Or in reverse:

Given the wavelength of the wave, the frequency is equal to the speed (of sound) in the medium divided by the wavelength.

In textbooks you can read: The speed of sound in fluids and solids is given by the square route of the compressibility modulus [in Pascal] divided by the density [in kg/m³]. As an indication: this results in a speed velocity of 1858 m/s for glycerin and 870 m/s for paraffin oil.

You can see that the wave propagation velocity in a medium, the acoustic vibration frequency and the corresponding wavelength have the following common basic relation: the wave propagation velocity equals the acoustic vibration frequency multiplied by the corresponding wavelength.

This relation is one of the fundamental corner stones of common wave propagation theory in physics.

For the definition of a wave you can than look in the Webster Dictionary.

Webster Dictionary Definition of a Wave.

Webster’s dictionary defines a wave as “a disturbance or variation that transfers energy progressively from point to point in a medium and that may take the form of an elastic deformation or of a variation of pressure, electric or magnetic intensity, electric potential, or temperature.”

Be aware that the equation in which is expressed that the wave velocity equals the frequency multiplied by the wavelength can easily lead to a completely erroneous interpretation.

For the physics in the equation you have to be aware about for example the following:

Measuring both the wavelength in the ‘wave’ evoked by the frequency stimulus and subsequently calculating the propagation speed of the ‘wave’ by multiplying wavelength with frequency has for example nothing to do with correct physics.

So, it is important to know: The speed of a sound wave that moves through a medium isn’t dependent on its frequency and its wavelength.

Because, the speed is given ...

Namely:

The speed of sound in fluids and solids is given by the square route of the compressibility modulus [in Pascal] divided by the density [in kg/m³]. As an indication: this results in a speed velocity of 1858 m/s for glycerin and 870 m/s for paraffin oil.

With this you can explore the above mentioned relationship.

And now we can look at: the content of the book
Applying physics makes auditory sense

• Introduction 10
• Objections against the traveling wave hypothesis 10

 

---

And we can look at:
Significance of the present findings for the concept of a traveling wave

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)].

---

It is in this sense that Békésy used the term ‘traveling wave’..." [pp. 511-513 of Wever et al. (1954)].

---

And we can look at:
Significance of the present findings for the concept of a traveling wave

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)].

---

And we can look at:
Ren’s unintentional attack on Von Békésy’s “Traveling Wave Theory”

An 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.

 

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.

Consequences:
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.

Conclusion:
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 f_r

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 f_r :

f < f_r : reduced in phase movement, with phase angle: 0
f = f_r : increase due to resonance but also a phase retardation with phase angle: ½ π
f > f_r : strongly reduced movement in opposite direction with phase angle: π

A demonstration in practice

Windows Media (WMV) is a video

Click on the video image above. Or by this button: ↓ Download  

 

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 f_r > f : move in phase with the stimulus.
That movement becomes larger if f_r 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 f_r < 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.

And now we can look at: the content of the book
Applying physics makes auditory sense

• Basilar membrane resonance phenomena instead of travelling waves 44

And we can look at: the figure 5 of the book
Applying physics makes auditory sense

 

An animation movie of it:

Deflection profiles of the basilar membrane around fc in sequential steps of T/12

An animation movie of it:

Windows Media (WMV) is a video

Click on the video image above. Or by this button: ↓ Download  

 

 

And before observe at first the way of movement of a singular resonator.

 

 

Video   Forced movements of a mass - spring system on a periodic stimulus.

( If you don’t see the video image below: refresh the page or clear your cache. )(video 4)

 

figure 5

Fig. 5. Deflection profiles of the basilar membrane around fc in sequential steps of T/12

 

 

---

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.

(complex function theory and conformal transformations) -- (deflection profiles basilar membrane)

Video Movement of the basilar membrane. By the Bernoulli effect.

( If you don’t see the video image below: refresh the page or clear your cache. )(video 2)

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.

---

Inventory:

Relationship between sound velocity, frequency and wavelength:
this is elementary physics and is covered in any physics textbook.
Velocity = frequency times wavelength.
This is particularly useful. You have relationships between them.

I can maybe add something about the interpretation of the equation.

Interpretation of the equation in which is expressed that the wave velocity equals the frequency multiplied by the wavelength [ in common wave propagation theory in physics ].

Take for example (sound):

With the travelling wave equation: the propagation velocity of the wave [in m/s] equals the frequency [in Hz] multiplied by the wavelength [in m].

This equation must be interpreted in the following way: the speed of a (sound) wave that moves through a medium isn’t dependent on its frequency and its wavelength.

The speed (of sound) - hence also the speed with which (sound) energy is transported - is a material constant and it therefore only depends on a number of properties of that medium. And the only way to change that speed is to change the properties of the medium.

Once the speed (of sound) in a medium is determined the above mentioned equation expresses the relation between the (sound) frequency and the wavelength.

The two have an inverse relationship.

• Given the frequency of the wave, the wavelength is equal to the speed (of sound) in the medium divided by the frequency.

Or in reverse:

• Given the wavelength of the wave, the frequency is equal to the speed (of sound) in the medium divided by the wavelength.

In textbooks you can read: The speed of sound in fluids and solids is given by the square route of the compressibility modulus [in Pascal] divided by the density [in kg/m³]. As an indication: this results in a speed velocity of 1858 m/s for glycerin and 870 m/s for paraffin oil.

You can see that the wave propagation velocity in a medium, the acoustic vibration frequency and the corresponding wavelength have the following common basic relation: the wave propagation velocity equals the acoustic vibration frequency multiplied by the corresponding wavelength.

This relation is one of the fundamental corner stones of common wave propagation theory in physics.

For the definition of a wave you can than look in the Webster Dictionary.

Webster Dictionary Definition of a Wave.

Webster's dictionary defines a wave as "a disturbance or variation that transfers energy progressively from point to point in a medium and that may take the form of an elastic deformation or of a variation of pressure, electric or magnetic intensity, electric potential, or temperature."

Be aware that the equation in which is expressed that the wave velocity equals the frequency multiplied by the wavelength can easily lead to a completely erroneous interpretation.

For the physics in the equation you have to be aware about for example the following:

Measuring both the wavelength in the ‘wave’ evoked by the frequency stimulus and subsequently calculating the propagation speed of the ‘wave’ by multiplying wavelength with frequency has for example nothing to do with correct physics.

So, it is important to know: The speed of a sound wave that moves through a medium isn’t dependent on its frequency and its wavelength.

Because, the speed is given ...

Namely:

The speed of sound in fluids and solids is given by the square route of the compressibility modulus [in Pascal] divided by the density [in kg/m3]. As an indication: this results in a speed velocity of 1858 m/s for glycerin and 870 m/s for paraffin oil.

We really must remind you to the fact that a mechanical vibration - and the sound stimulus is such a vibration - in a fluid, or in this case water like perilymph, will always propagate with the speed of sound, which has typically here the value of 1500 m/s.

That is just one of those constraints dictated by general physics. And with the equation that counts for the relation between sound velocity, frequency and wavelength we simply can calculate that for a 1000 Hz stimulus the corresponding wavelength in the perilymph is 1.5 meter. So approximately equal to 50 times the length of the active partition of the basilar membrane.

That is the only reason why the round window is moving in opposite direction related to the oval window. A widely reported always observed phenomenon in experiments.

And under the existing conditions in the cochlea there is no physics ground for so-called ‘slow waves’ with wavelengths even in the order of fractions of a millimeter. With the same equation for the relation between wave propagation velocity, frequency and wavelength as is used for the ‘fast’ running waves here above.

Just because such a slow wave demands the propagation of a row of successively higher and lower pressure areas with sizes in the order of those wavelengths and even smaller. And that is impossible in general physics. The incompressibility of the perilymph fluid makes this impossible.

It cannot be that a mathematical wish for explaining the existence of a hypothesized traveling wave with a small wavelength prescribes that physics has to offer the possibility for such a slow wave. Just because the general physics rule prescribes that wave propagation velocity equals frequency times wavelength.

And therefore the only possibility that remains is that under the incompressibility constraint the entire perilymph fluid column between oval window - helicotrema - round window is moving as a whole, while it is stimulated by a mechanical vibration of the stirrup.

If we look closer to the basilar membrane properties, we observe that there exists a frequency-place related distributed resonance capability. With a subdivision that has a logarithmical scale from apex to base. High resonance frequencies near the base and low resonance frequencies near the apex.

Actually this unique property is the cause that a stimulus, that is equally present all over the length of the basilar membrane, evokes phase related movements which appear as a ‘wave’ that is always running from base to apex.
And it is this ‘wave’ phenomenon that is erroneously interpreted as the ‘traveling wave’ that transfers the sound stimulus.

And of course the perilymph fluid can be stimulated from both sides. Wever and Lawrence have reported that already in 1950. They reported that stimulating either the oval window or the round window results in identical cochlear potentials.
But that doesn’t imply that there have to run traveling waves in both directions.
We can only draw the conclusion that a perilymph push-pull caused by a sound stimulus isn’t dependent on the pathway that is chosen.

But even more, Wever and Lawrence have reported a lot more already in 1950, but we all need to read the article again with regard to a nowadays mathematical technique with vectors, even what Wever and Lawrence have simple reported in 1950, because Wever and Lawrence talk about "a sort of vector summation" and they have said not much wrong with it.

What they have reported in 1950 can nowadays be demonstrated with the nowadays mathematical technique with vectors, the demonstration 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. And we can see its consequences....























The final conclusion of the view about these experiments excecuted by Wever and Lawrence in 1950, and what they have reported, 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. A direct measurement of velocity inside the cochlea is known as extremely difficult. So far every attempt fails, mostly because of the intolerable disturbances of the properties in the location that has to be examined. This makes the experimental results unreliable. And non-invasive measurements still show not enough details of fluid movements.

But from what really happens there we nevertheless can still make a reliable imagination, which is simply based on physics and the physiological properties and parameters which exist in the cochlea.

Let us make an inventory of them.

The physiological structure data:

• The walls of the cochlear envelope are extremely rigid. Hardly compliant to not compliant at all. So bone conduction based on deformation of that envelope is not possible.
  

• The cochlea shaped cavity is subdivided into the perilymph filled duct, which is folded at the apex and which parts are the scala vestibuli between oval window and apex and the scala tympani between apex and round window.

• In between these two scalae the third one - scala media - is located, filled with endolymph.
  

• The partition between scala vestibuli and scala media is formed by the Reissner membrane. This membrane is extremely thin, but on all available electron microscope pictures it is observed as straight, except for Ménière cochlea, where it is curved into the direction of the scala vestibuli. This membrane has compliance.
  

• The partition between the scala tympani and the scala media is formed by the basilar membrane. This membrane has substantially more volume, while both the inner and the outer hair cells are embedded in it. This membrane has a place located frequency dependent compliance. It can be observed as a resonance device.
  

• The hair bundles of the outer hair cells are at their top connected with the tectorial membrane, a rim structure that is completely located in the scala media and connected with the bony envelope of the cochlea.
  

• At every location along the cochlear partition the cross sections of scala tympani and scala vestibuli are practically equal in size. In average the channel diameter is 0.3 mm.
  

• There exists a tapered shape in the perilymph duct, larger at the base to smaller at the apex.
  

• The maximal deflections of the oval window and the round window are estimated to be in the order of a few micrometers.
  

• Deflections of the basilar membrane also do not exceed a few micrometers. Otherwise the hair bundles of the outer hair cells would be damaged due to overstressing.
  

The involved material quantities:

• The perilymph fluid in the scala vestibuli and scala tympani as well as the endolymph fluid in the scala media has a density equal to that of water. So 1000 kg/m³.
  

• Both fluids are incompressible and have a low viscosity, comparable with water. This can be considered in practice as ‘viscous free’.
  

• The propagation velocity of acoustic vibrations in the perilymph is 1500 m/s.
  

Cardinal fluid dynamics numbers:

• The main criterion for non-turbulent fluid flow in the cochlea is the Reynolds number. Calculation in case of the highest hearable frequency stimulus of a 20 kHz vibration with an amplitude of 1 micrometer with a dynamic viscosity coefficient equal to that of water: 0.001 Ns/m² gives for this Reynolds number a value of 36. For vibrations with lower frequencies and with similar amplitudes this Reynolds number is proportionally lower.
  

Hence the Reynolds number is for all situations far below the boundary of 2000, the value that counts for upper boundary of the laminar flow conditions. And therefore the perilymph flow inside the cochlea is definitely non-turbulent.

All these aspects together result in the fact that it is allowed to consider the perilymph movement inside the cochlea as a periodic movement that can be theoretically expressed by the non-stationary Bernoulli equation.

---

For those of you who think that I (Heerens) misuse the Navier-Stokes theory in the case of the cochlear fluid dynamics I can point to the following elucidating introductory explanation placed on-line on the Internet by the Academic Medical Center of Amsterdam that I (Heerens) cite here:

-- The Navier-Stokes equations are a set of equations that describe the motion of fluids (liquids and gases, and even solids of geological sizes and time-scales). These equations establish that changes in momentum (acceleration) of the particles of a fluid are simply the product of changes in pressure and dissipative viscous forces (friction) acting inside the fluid. These viscous forces originate in molecular interactions and dictate how sticky (viscous) a fluid is. Thus, the Navier-Stokes equations are a dynamical statement of the balance of forces acting at any given region of the fluid.

They are one of the most useful sets of equations because they describe the physics of a large number of phenomena of academic and economic interest. They are useful to model weather, ocean currents (climate), water flow in a pipe, motion of stars inside a galaxy, flow around a wing of an aircraft. They are also used in the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of the effects of pollution, etc.

The Navier-Stokes equations are partial differential equations which describe the motion of a fluid, so they focus on the rates of change or fluxes of these quantities. In mathematical terms these rates correspond to their derivatives. Thus, the Navier-Stokes for the simplest case of an ideal fluid (i.e. incompressible) with zero viscosity states that acceleration (the rate of change of velocity) is proportional to the derivative of internal pressure. Poiseuille’s Law and Bernoulli’s equation are special cases of 1D Navier-Stokes.

The fluid motion is described in 3-D space, and densities and viscosities may be different for the 3 dimensions, may vary in space and time. Since the flow can be laminar as well as turbulent, the mathematics to describe the system is highly complex.

In practice only the simplest cases can be solved and their exact solution is known. These cases often involve non turbulent flow in steady state (flow does not change with time) in which the viscosity of the fluid is large or its velocity is small (small Reynolds number).

For more complex situations, solution of the Navier-Stokes equations must be found with the help of numeric computer models, here called computational fluid dynamics. Even though turbulence is an everyday experience it is extremely hard to find solutions for this class of problems. Often analytic solutions cannot be found.

Reducing the models to 1D, as is often done in fluid dynamics of blood vessels, makes the problem handsome. --

[See also: http://onderwijs1.amc.nl/medfysica/compendiumDT.htm edited by N. A. M. Schellart 2005]

You can see for yourself that I (Heerens) have derived the analytical solution for the non-stationary non-viscous incompressible time dependent wiggle-waggle movements directed along the core of the perilymph duct. Because in that case the reduction of the complex set of Navier-Stokes equations to the non-stationary Bernoulli equation is fully permitted.

And this finally results in the fact that everywhere inside the perilymph duct the evoked pressure variations are proportional to the sound energy stimulus.

This means that by resonance in the basilar membrane, i.e. the frequency-place related distributed resonance capability, the stimulus can evoke simultaneously all the frequency contributions of the sound energy signal, including the exact phase relation for each contribution, which will be sent to the auditory cortex.

---

Nonstationary Bernoulli

 

 

---

---

---

---

Perhaps I could share some idea for further research.

If we could make actual and correct pressure measurements in the cochlea to reveal wether the non-stationary Bernoulli effect is a good description of the actual physics-of-how-the-cochlea-isolates-frequencies-along-its-length?

Organ of corti operation. Inner hair cells are the leftmost row, outer hair cells are the other three rows.

I would consider:

I would propose to use a pitot tube, with sensor in the side wall [ B in the next figure, left side in that figure ] to actual have correct pressure measurements in the perilymph flow tube inside the cochlea.

Therefore I would propose to use a pitot tube, with sensor in the side wall [ B in the figure, left side in the figure ] to actual have correct pressure measurements in the perilymph flow tube inside the cochlea.

So, I would propose to use a pitot tube, with sensor in the side wall [ B in the figure, left side in the figure ]

---

for a good mutual understanding
our new hearing paradigm
the theory we have described in our booklet is utterly non-linear.

has fundamental and cardinal a number of very important consequences.

---

Modification of the Fletcher-Munson curve
based on dB[SPL] scale into the dB[SEL] scale
by a -6 dB/octave correction.

This forms the basis for the appreciation of 1/f sound compositions.

From limited knowledge, decades ago, dating back to the nineteenth century:

Ohm's law of specific acoustic energies was the first biological application of Fourier's theorem.

Actually, it was already suggested in J. Müller.s Handbuch der Physiologie des Menschen, Vol. II, Hölscher, Coblenz, 1838.

At that time there was no alternative but to think in terms of energy and mechanics rather than information and neurons.

The human was considered a machine.

Our research interest concerns the verification on correct application of physics in explanations and hypotheses about hearing related processes.

Our research gives us many indications, confirms the suspicion that with regarding auditory sense, we really again go all the way back to the analysis of the sound energy, as was already suggested by Ohm decades ago,

by discovering 'The non stationary Bernoulli effect' inside the cochlea, actually, suggested in 2010: that there is: "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."

Of course, the term acoustic energy is justified until the hair cells act (by the biological particulars of the human ear).

Correspondingly, one can check by physics to what extent inherited terminology like „acoustic energy" is still appropriate to the defenition of acoustic energy, that the sound pressure is squared and there the logarithm is taken.

And put a corresponding investigation into question:

considering 'the term acoustic energy is justified until the hair cells act', is it then not completely overlooked the fact that the true physical value of the acoustics energy is proportional to both the square of the amplitude and to the square of the frequency?

After all, is it not that what explains the 1/f character of pleasant music?

The differentiation step in that transfer makes that all frequency contributions in the sound pressure signal with 1/f amplitude ratio's have equal contributions in the sound energy signal.

1/f spectra have the unique distinction of being "scale invariant" in the sense that the energy in an interval df is proportional to df.

The 1/f spectra in fact have the property that the in an interval with width df available energy is proportional to df but not with f. There, namely "scale invariant" attribute for. It is not the energy, but the signal amplitude with which 1/f scales.

Also for the sound spectrum usually offered to our hearing, it is like that. And it turns out that there are very many natural sources to have a 1/f spectrum distribution.

And then it is also consistent with the fact that each frequency interval of width df [difference in Hz] at a random location in the spectrum with frequency f [value in Hz] has an energy content that depends on the value of the frequency interval df but does not depend on the frequency f itself.

I have heard it said on a number of occasions that 1/f spectra are very commonly encountered among natural signals, and one might perhaps expect the auditory system to reflect this fact in its design.

In nature it is quite normal that there are noises generated by noise sources, and the noise spectral energy density - the sound energy per frequency band with constant width - is constant.

But that also means that the sound signals provided by the sources themselves consist of frequency fees, which have amplitudes inversely proportional to their frequency f. Which leads to the so-called 1/f spectrum.

If the hearing of mammals differentiates - so there out of the sound pressure signal per frequency contribution f also an f time greater perilymph velocity, which neutralizes the existing 1/f factor therein - and then squaring, so that the final sound energy signal anywhere on the basilar membrane therefore no longer depends on the frequency f, the auditory sense is in the best possible way adapted to the perception of sound sources with a 1/f spectrum.

Looked at classroom and offices, respectively, and generally found that the spectrum of summed background sounds rolls off (declines in amplitude) according to a 1/f function, somewhat similar to pink noise.

That's because both natural sources, but also the most musical instruments inside the generated energy with constant spectral energy density - so 1/f spectra exciting - are broadcasting periodic noises.

Of course 1/f rules: not unlimited in long-term spectra. But there does not exist in nature normally indefinitely sounding by sound sources.

1/f, a reasonable approximation of the occurring spectral energy density.

And now that Bernoulli effect inside the cochlea: differentiating and quadrating.

From here, we can make use of the so-called 1/f relation for sounds found in nature. By this 1/f relation, the sound pressure amplitude p0i of a pure tone in a tone complex will be reciprocal to its frequency fi . Immediately, the reason for the preference for 1/f sound contributions becomes clear: The signal strength of each stimulus contribution on the BM (basilar membrane) becomes frequency independent. Not surprisingly, this well-established 1/f quality of sounds is a phenomenon that is omni-present in nature. The mammalian auditory sense shows a perfect adaptation to such sounds.

Sometimes: that 1/f character is just globally. It makes sense to me because nature is never exact in every detail.

But sure: there is a lot of 1/f associated with in nature, the speech and music sounds. And one wonders whether it affects the functioning of the auditory organ. But for me: It looks now: Or is it not more the other way around: the mammalian auditory sense shows a perfect adaptation to such 1/f sounds.

When a theory suggests that our hearing differentiates and squares. And that then the brains, of the most obvious, is getting evenly distributed sound energy frequency signals. So it benefits from that the noise amplitude spectrum is of the nature of 1/f.

In fact: the occurrence of 1/f spectra and hearing then show a logical duality. And then: Considering 'the term acoustic energy is justified until the hair cells act', Yes: the true physical value of the acoustics energy is then proportional to both the square of the amplitude and to the square of the frequency, if one considers that the mammalian cochlea differentiates and squares the incoming sound pressure signal inside the cochlea itself, in terms of physics, totally hydrodynamic in origin, contrary to that an early neural mechanism is responsible for it, contrary to brain processes it. Totally hydrodynamic in origin! Non stationary Bernoulli effect inside only the cochlea.

So now, to take a position about a terminology of acoustics energy:

Consequences of this for audiologic research:

Let's look at:

The Fletcher-Munson curve:

This curve expresses the data for the sensitivity of the human hearing sense.

It shows the hearing threshold based on the sound pressure level dB scale [dB SPL].

In an equation this quantity is expressed as:

The Fletcher-Munson curve in a graph:

This curve shows a very remarkable effect.

It is only flat in the frequency region between 3 and 4 kHz.

Beyond the frequency domain important for humans.

In a transfer from the sound pressure stimulus into the sound energy stimulus, the relation between the sound pressure quantity dB(SPL) and sound energy quantity dB(SEL) can be calculated as:

The final result is given by:

Or between 'sound energy level' and 'sound pressure level':

And this means that from dB(SPL) to dB(SEL) we have to correct the Fletcher-Munson curve with a subtraction of 6dB/octave:

In the graph this is given by:

Correction with 6 dB/octave gives:

The sound energy sensitivity curve.

And now the curve shows the sensitivity for sound energy contributions:

Conclusions:

This audiology phenomena is explained in a way when we follow the paradigm in which the cochlea analyzes the sound energy frequency spectrum. In terms of physics, totally hydrodynamic in origin.

Here we have explained:

Modification of the Fletcher-Munson curve based on dB[SPL] scale into the dB[SEL] scale by a -6 dB/octave correction.

This all together forms the basis for the appreciation of 1 / f sound compositions.

In the hearing world and the sound recording world - based on auditory - one makes use of the so-called dB [SPL] [Sound Pressure Level] values.

These are then found by making use of the so-called "least squares" [Root-Mean-square] average calculation.

See:

http://en.wikipedia.org/wiki/Acoustic_pressure#Sound_pressure_level and:

http://en.wikipedia.org/wiki/Root-mean-square

And then systematically - if one does not consider a differentiating and squaring process in the cohlea - one makes only use of the Fletcher-Munson curve.

Sensitivity curve [Fletcher-Munson curve] is a curve for normal hearing sense.

In fact if you look at the definition of dB SPL, which stands for "decibel sound pressure level", to calculate realy everyting all right, using those dB SPL calculations, one can now look further also at the influence of the frequency thereto. If you do so, you'll see that with a same soundpressure at 2000 Hz the dB SPL value compared to that one of the 1000 Hz, is a factor 4, i.e. 6 dB higher than with using the existing definition of the dB SPL to calculate the value.

That's because it overlooks the fact that with the same pressure amplitude in each volume the same mass present at 2000 Hz moves two times rapidly as at 1000 Hz.

And then the energy of motion is still proportional to the square of the velocity.

The dB SPL scale should therefore be corrected - 6 dB ear sensitivity subtracted per octave.

Which means that for each octave there a 6 dB must be subtracted.

And that it is equally again connected with the Fletcher-Munson curve or isophone zero dB, the correction curve in use by audiometry.

If you apply there the 6 dB per octave correction, you see suddenly that most of that remarkable notability of the auditory sensitivity to rising pitch in the most sensitive field of our hearing disappears.

That means it is mutually so closely related.

This all together forms the basis for the appreciation of 1 / f sound compositions.

Video Modification of the Fletcher-Munson curve based on dB[SPL] scale into the dB[SEL] scale by a -6 dB/octave correction.

( If you don’t see the video image below: refresh the page or clear your cache. )(video 6)

---