sequentially-scrolling-Nonstationary Bernoulli sequentially-scrolling-Nonstationary Bernoulli
An animation movie of it:
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)].
by Willem Christiaan Heerens
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.
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
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.
“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
And we can add the following to it:
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.
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.
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