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Re: A new paradigm?(On pitch and periodicity (was "correction to post"))

Dear Peter van Hengel and List,

Peter, my apologies for not responding to your comments earlier but it has 
cost me a lot of time to react in a well-considered way, supported by 
careful explanations.
And to the members of the AUDITORY List: my apologies for the length of it.

But first I would like to congratulate you, Peter, with the textbook 
written by you in co-authorship with your former promotor prof. dr. ir. H. 
Duifhuis that was announced to appear on September  1  2011 under the title:

Cochlear Mechanics

Authors:  Hendrikus Duifhuis & Pieter Willem Jan Van Hengel 
250 pages | Springer-Verlag New York Inc. | 2011 | September 2011 

ISBN-10: 144196116X    &   ISBN-13: 978-1441961167 

A brief summary is also given on the Internet and I cite:

** The field of cochlear mechanics has received an increasing interest over 
the last few decades. In the majority of these studies the researchers use 
linear systems analysis or linear approximations of the nonlinear (NL) 
systems. Even though it has been clear that the intact cochlea operates 
nonlinearly, lack of tools for proper nonlinear analysis, and widely 
available tools for linear analysis still lead to inefficient and probably 
incorrect interpretation of the biophysics of the cochlea. An example is 
the presumption that a change in cochlear stiffness (at hair cell level) 
must account for the observed change in tuning (or frequency mapping) due 
(e.g.) prestin application. Misconceptions like this need to be addressed 
in a tutorial that is lucid enough to explain basic differences between 
linear and nonlinear cochleae at a graduate level. This book presents a 
useful and mathematically justified/justifiable approach in the main part 
of the text, an approach that will be elucidated with clear examples. The 
book will be useful to scientists in auditory neuroscience, as well as 
graduate students in biophysics/biomedical engineering. **

I have cited this summary of your textbook because we both have statements 
in common, however on one cardinal issue I see that you and I strongly 
disagree. Actually even almost to the point of insult regarding the other’s 
scientific skills.

I am pleased to read that you also reject the hypothesis about the role of 

I also want to remark that you and I agree that the intact cochlea operates 
nonlinear and that at the moment tools for proper nonlinear behavior are 
lacking and the linear tools for analysis lead to inefficient and probably 
incorrect interpretation of the biophysics of the cochlea.

Hence we both opt for at least a quadratic functioning of the cochlea. 
Because this is in general the first next step in non-linear operation.

But now the subject we seriously disagree about.

Let me first give a reply on the first item of your supporting comment for 
Dick Lyon:

You wrote:

** As I have stated before fluid physics states that a fluid domain (such 
as the cochlea or a pond) with a flexible boundary subject to a restoring 
force (such as the cochlear partition or the pond surface) MUST 
exhibit 'ripples' on the surface. **

I wonder what makes you so sure about that statement. And I elucidate this.

Ripples on the surface of a pond are actually a combination of transverse 
and longitudinal waves; therefore, the points on the surface follow orbital 
paths. What actually aren’t my words, but the explanation of this wave 
phenomenon you can find in many textbooks. 
You can observe such ripples that start and spread circularly after you 
have thrown a pebble in the pond. That the water particles in the interface 
layer follow orbital paths you can observe as well if in advance you have 
sprinkled small cork particles over the area where the ripples can be 
expected. You clearly see that these particles not only move with the water 
surface in vertical direction, so in line with the restoring force, but 
also in lateral direction, however in time average they remain at their 
lateral location.

But now the following issue: let me name it ‘oil on the waves’. The ripples 
on the pond, we have observed before, are becoming subject of a relative 
strong damping in case oil is spread as a thin layer on that water surface. 
Whatever the explanation in detail for this damping phenomenon may be, it 
is anyhow clear that the interface conditions are changed substantially by 
that thin oil layer and thus the constraints for evoking ripples are 
changed substantially as well. 

Now you can wonder what will happen if it isn’t a thin oil layer that is 
drifting on the water surface, but a piece of thicker foil with a much more 
internally connected structure. I can guarantee you, and by doing that 
experiment you can verify my statement, that the ripples that will pass 
along that foil will spread according to the wave propagation rules of 
Huygens and the edge of that foil piece will act as the origin of a 
secondary wave front. But the inner part of the foil remains to a high 
extend free of wave motions. And keep in mind that the restoring force – 
the gravitational force – is still present. 

And now the last step in my thoughts about evoking ripples: what will 
happen if we substitute for the air on top of that foil another fluid with 
almost identical qualities as the water below the foil? There won’t be a 
ripple left at all. All the conditions necessary for the ripple phenomenon 
do not exist anymore.

So I have a question for you: 

Isn’t that a valid reason for my doubts about your statement that a fluid 
domain under the condition like you have formulated above, with the only 
condition that a flexible boundary is subject to a restoring force MUST 
exhibit 'ripples' on the surface? 

You can ask me the question: on what grounds are you arguing? Do you have 
the skills for this scientific subject? 
Well let me inform you that all the above mentioned boundary interface 
issues were part of my research that lead to my masters degree in 1967. For 
that research project in that time I had to follow the course called ‘Hydro-
dynamical Engineering’. A course really loaded with the equations of Gauss, 
Green and Navier-Stokes and even more. But also solutions of Laplace’s 
equation in case of potential flow. 
Later in my career that same knowledge, together with a post graduate study 
of ‘Special Function Theory’, made it possible for me to publish a series 
of papers about the analytical solution of Laplace’s equation in toroids 
and cylinders with several types of boundary conditions. 

I am informed that you have a masters degree in mathematics, not in 
physics, so if you are interested, I can give you full references, or even 
scanned copies of those papers. These analytical mathematical solutions can 
also play a role in achieving reliable rules of  thumb  for instance in the 
case of dimensioning potential distributions in cochlear implant research.
And finally it was also part of the basis for the academic courses I gave a 
number of years with subjects like ‘Introduction to Electricity and 

Is all this detailed CV information for you satisfactory enough to take my 
contributions in this subject serious?

I have followed the indication you gave me in the beginning of your reply 
to me:

** I hesitate to get involved in this discussion as I have tried to explain 
the hydrodynamics behind (transmission line) cochlea models before in 
another thread on this list and don't like repeating myself. **

So I found out that you are familiar with this subject. You explicated that 
in the discussion thread of March 2010:

** In response to Martin Braun I have a question how there can be a motion 
of fluid without a pressure change, or a sound wave without a pressure 
change? The fluid mechanics I know (Bernoulli etc) states that any motion 
of fluid is always linked to a pressure gradient, therefore pressure 
difference, and I thought a sound wave was defined as pressure differences 
traveling. **

Correct me if I interpret your explanation wrong: do you really mean 
Bernoulli related to a pressure gradient? And then a gradient in the 
direction of the flow?

About the pressure gradient you are right, but only in the case of a 
viscous flow. Because in that case the internal friction forces result in a 
pressure gradient in the flow direction. For a tube with circular cross 
section this negative gradient – the pressure drop per unit of length – is 
proportional to the coefficient of viscosity multiplied by the volume flow 
and divided by the fourth power of the tube diameter. In  hydrodynamics 
this is well known under the name: ‘flow relation of Hagen-Poiseuille’. And 
this relation is valid for a laminar flow in the tube, which exists if the 
Reynolds number – the ratio between inertia forces and friction forces – is 
less than 2000.

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 
And as you can see in my attachment:  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. 
There doesn’t exist a pressure gradient inside that fluid column, as long 
as the area of the cross section remains the same. If the diameter of the 
cross section changes as function of the distance measured along the tube 
axis, while the shape of the cross section remains the same, the fluid 
velocity will be reciprocal to the squared diameter and consequently the 
local pressure decrease changes with the fourth power of that diameter. So 
if the duct has a tapered shape the pressure distribution along the duct 
shows a small pressure drop in the wider areas and an increasing pressure 
drop in narrower areas. However that isn’t the pressure gradient that acts 
as the driving force you have in mind, because this pressure distribution 
isn’t the cause of the flow in the fluid but the result of it. 

In that same contribution in March 2010 you stated:

** All I have said about the traveling wave was based on a fluid with a 
flexible boundary which has a restoring force. If the fluid is moved, in 
this case you will always get a traveling wave. **

And exactly this statement I do not only call in question, I even reject 
this traveling wave hypothesis. It isn’t a law of Medes and Persians that 
you always get a traveling wave in case the fluid moves in presence of a 
restoring force in the wall of the involved duct. 

Referring to the extensive research on this topic by Shankar and Kumaran [1 
& 2] and making an indicative calculation example in case of a 2 micrometer 
total displacement of the oval window on a 20 kHz pure tone signal, we get 
as a result for the maximum perilymph velocity 0.012 m/s. And with a 20 
micrometer oval window displacement  on a 1 kHz pure tone this maximum 
perilymph velocity is 0.006 m/s.
These velocities are far too low to generate periodic instabilities that 
can lead to what can be observed  as a traveling wave. Neither in the fluid 
nor in the wall. 

Just follow a while Albert Einstein – a real wise man – with his 
quote: "Imagination is more important than knowledge". 

So please use your imagination. And then, are you really convinced that in 
a fluid column that is wiggling in its length direction with hardly 
observable displacements – and if so even with serious measuring problems, 
even with a high resolution microscope – that wiggling will generate 
pressure disturbances in the size of a few millimeters which propagate in 
that fluid column? Just because your mathematical model prescribes these 

See also the references:

1. Shankar V, Kumaran V. (1999) Stability of non-parabolic flow in a 
flexible tube. J. Fluid Mech. 395: 211- 236. 

2. Shankar V, Kumaran V. (2000) Stability of fluid flow in a flexible tube 
to non-axisymmetric disturbances. J. Fluid Mech. 407: 291- 314. 

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 that I give in the 

For me actually this was both a renewed confrontation with the knowledge 
that I had gathered now 46 years ago and the joyful experience that I 
didn’t lost my touch on hydrodynamics.  

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 my reply to Dick Lyon of Monday the 19th  of September 2011  I suggested 
the experiments with the composed beat series of seven pure tones, based on 
the 10 kHz frequency; 4 Hz spacing in frequency; a 0.0625 Hz detuning on 
the 2nd  4th and 6th  frequencies and all sine or alternating sine cosine 

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.

And now I want to address both a comment and a question to you about your 
provocative statement:

**In the cochlea these are refered to as traveling waves. The wave energy 
is not traveling in the boundary itself but in the fluid. Any attempts to 
prove that such waves do not exist, or are based on 'bad physics', are 
unfortunately based on a lack of understanding of the fluid mechanics.**

My comment:

The non-stationary potential flow according to Bernoulli in the perilymph 
duct, like I have 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 
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 I gave in the 10 kHz example.

And now my question to you:
Are you still fully convinced that my contribution must be categorized as 
the next fruitless attempt of someone who thinks he can contribute on your 
high scientific level? 

Since there hasn’t appeared a single reaction about it on this AUDITORY 
List, I really wonder if there is anybody on this List who has done the 
experiments that I have described and who has verified the results 
belonging to them for satisfying his/her curiosity. I have more than enough 
testimonies from independent examiners who confirm my claims. But I hardly 
can imagine that none of the auditory experts can’t hear them.

What for instance is observed from the following combinations of 

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 

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

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

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 me add another 
experiment with which this is absolutely impossible. 

Let us chose the following frequency contributions:

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. 

I cannot believe that these salient auditory perception results for each 
chosen average frequency over the entire auditory frequency domain are 
heard by everybody else except the auditory experts, being the members of 
this AUDITORY List.

Neither I can believe that the experiments mentioned above fit well in the 
transmission line concept, of which you are an adept.

Peter you wrote:
** Whether the traveling wave is the only mechanism responsible for 
transporting sound energy to the hair cells is still a valid question, but 
untill an alternative model produces similar or better results on modeling 
physiological, pshychophysical and OAE date, I'll stick with the 
transmission line. **

Of course you can. Probably I would have done the same if I were as 
convinced as you about the transmission line calculation model that you are 
applying. And what then counts is that such calculation models must be in 
agreement with the rules and laws of physics. 

You also replied with:
** Things like pitch perception and the missing fundamental can perhaps not 
be explained purely by looking at the average excitation caused by the 
traveling wave, but I don't think anyone ever claimed they could. **

No indeed, but do you acknowledge that it is strange  indeed that we have 
to apply a relative great number of different models with in many cases 
significantly restricted validity domains for the description of all the 
hearing phenomena?  And yet there are still a substantial number of 
phenomena that cannot be explained or that are falling in the category 

In all widely explored fields of science scientists always strive for a 
consistent theoretical structure that covers the concerned scientific field 
as much as possible as a whole. And all the scientific contributions in the 
auditory field, I have studied so far, give me the impression that this is 
also the goal of all the serious workers in auditory science. 

Therefore I don't quite understand your last remark in the support of Dick 

**In my opinion it is good to develop new theories, but we should attempt 
to integrate them with existing ones instead of throwing away something 
that has proven to work.**

That’s the way science can develop according to Karl Popper. Small steps in 
a row.

Let me give you some examples to show you what the consequences of your 
statement imply. 

A scientific revolution was necessary to step over from the geocentric 
world view to the Copernican heliocentric world view, since both theories 
are incommensurable.

And Einstein’s theory of relativity cannot be integrated in the physics 
theory of Newton. 

And there are many, many more examples like these to give. 

If the constraint for new scientific work would be dictated as you suggest: 
restricted to investigations that fit within the existing knowledge, I fear 
that scientific progress will become very poor.     

In your reply to Matt Flax you wrote:

** just to be clear: I only used my pond analogy to indicate that the 
traveling wave observed on the surface is caused by the traveling wave in 
the fluid. In both cases there is a fluid domain with a flexible boundary 
with a restoring force. I never meant to imply that surface waves on a pond 
can be used to describe the details of what's going on in the cochlea. The 
physical processes and forces at work in the two cases are completely 
different. Sorry if this was confusing. ** 

Well now you are arguing in the same way as Nobel prize laureate Richard 
Feynman did. He warned that using the ripples in a pond as example for 
other wavy phenomena is mostly wrong, because of the unnoticed complexity 
in this water wave phenomenon.

And then on the comment of Matt, I cite here:

++ I have no problem with the passive travelling wave, however with respect 
to actively induced movements, the latest experimental data shows that 
there are ONLY forward travelling waves (check Ren's experiments for 
example). ++

You replied to him:

** thanks for the support. One small comment to your statement that there 
are only forward traveling waves. I think much of the confusion stems from 
the fact that the observed motion of the cochlear partition is often 
referred to as the traveling wave. In actual fact this is only 
a 'reflection' of the actual wave which is traveling in the fluid. The 
fluid supports waves traveling in both directions, but the unique 
properties of the cochlea cause only the appearance of a forward traveling 
on the cochlear partition.**

Yes that is commonly supposed to happen. However experimental data of 
Tianying Ren – at first flawed by other cochlear scientists – but quite 
recently confirmed by experiments with a different experimental method 
under the guidance of De Boer and Nuttall, show that no backwards traveling 
waves can be found. 
After that surprising discovery De Boer expressed in public that this was a 
clear anomaly and this problem should be solved in order to repair the 
existing OAE theory that uses the backward traveling wave concept and he 
and his coworkers have started an investigation with a concept that 
hypothesizes, that evoked forward traveling waves in the cochlear duct will 
transfer the OAE stimulus from the scala tympani to the eardrum.    


Ren T. (2004) Reverse propagation of sound in the gerbil cochlea. Nature 
Neuroscience 7, 333 - 334

De Boer E, Zheng J, Porsov E, Nuttall AL. (2008) Inverted direction of wave 
(IDWP) in the cochlea. JASA 123(3):1513-1521.

He W, Fridberger A, Porsov E, Ren T. (2010) Fast reverse propagation of 
sound in the living cochlea. Biophys J. Jun 2;98(11): 2497-505.

Your reply to Matt:

** If one wants to observe the reverse traveling waves in the cochlea it is 
necessary to measure fluid velocity, which I believe is not yet possible. 
Model calculations clearly show the reverse traveling wave and produce 
results in accordance with data on OAEs (see e.g. the work of Mauermann et 
al or Epp et al). But I'll (re)check the work of Ren to make sure I'm not 
relying only on what I believe to be true ;-).**

Yes and that is precisely the problem: ‘model calculations clearly show the 
reverse traveling wave’. But in reliable experiments in practice, [ Ren, De 
Boer et.al. ] they aren’t observed. Only forward traveling waves along the 
basilar membrane are found. 

In the summary of the 3rd  mentioned paper [He, Fridberger, Porsov and Ren] 
there are given a number of interesting conclusions:  

-- … However, the fundamental question of how the otoacoustic emission 
exits the cochlea remains unanswered. In this study, emissions were 
provoked by two tones with a constant frequency ratio, and measured as 
vibrations at the basilar membrane and at the stapes, and as sound pressure 
in the ear canal. 
The propagation direction and delay of the emission were determined by 
measuring the phase difference between basilar membrane and stapes 
These measurements show that cochlea-generated sound arrives at the stapes 
earlier than at the measured basilar membrane location. Data also show that 
basilar membrane vibration at the emission frequency is similar to that 
evoked by external tones. These results conflict with the backward-
traveling-wave theory and suggest that at low and intermediate sound 
levels, the emission exits the cochlea predominantly through the cochlear 
fluids. …--

Because both the sound related vibrations of the stapes apparently started 
earlier than the corresponding basilar membrane movements and the basilar 
membrane movements appear to be similar to movements evoked by external 
tones, we might also draw the conclusion that the otoacoustic signal is 
generated in the stapes and then as a normal sound signal transferred to 
the basilar membrane.

Let us be wise. The physics society is already thrown into commotion by the 
news that CERN announced the measurement of neutrino’s running faster than 
light. We mustn’t add the information that acoustic vibrations even travel 
back in time, because the OAE’s evoked in the cochlea are reaching the 
stapes earlier than the location from where they are assumed to come.

This divergence between predicting models and reliable verifiable 
experiments shows that the traveling wave model might be seriously wrong.

Finally I will end with the quote of sir James Lighthill, that forms the 
beginning of the Memorial Tribute dedicated to him and written by Lokenath 
Debnath in 1998, the year Lighthill passed away.

[See also: http://www.emis.de/journals/HOA/IJMMS/22/4667.pdf ] 

“... as Sir Cyril Hinshelwood has observed ... fluid dynamicists were 
divided into hydraulic engineers who observed things that could not be 
explained and mathematicians who explained things that could not be 

James Lighthill

This because it expresses my impression very well for the moment.

Kind regards,

Pim Heerens