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The relationship between frequency and wavelength

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

sequentially-scrolling-book-page010

 

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.

459-for-the-short-traveling-wave

 

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 fr

Let us observe the reaction of a spring-mass-system
on a periodic stimilus

459-spring-mass-system-mass-spring-pointofimpactstimulus

 

If the resonator is brought into a vibrating movement, then three different situations can exist, dependent on the relationship between stimulus frequency f and resonance frequency fr :

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

A demonstration in practice

Windows Media (WMV) is a video

459-spring-mass-system-mass-spring-pointofimpactstimulus

 

Followed by the remarkable mechanical setup of the basilar membrane:
This basilar membrane [ BM ] exists of an array of small resonators, that have gradually decreasing resonance frequencies from the round window up to the helicotrema.

And then in case of an everywhere equal in phase stimulus on the entire BM, the following is happening:

All parts of the BM having fr > f : move in phase with the stimulus.
That movement becomes larger if fr approaches f closer and will retard gradually in phase.

In case of resonance a large movement is and there exist a phase retardation of ½ π
All parts of the BM with fr < f are more and more moving in opposite phase with the stimulus and with a growing decreasing in deflection.

And what phenomenon is comparable to this?
The “wave” in the stadium!

And dependent on the quality factor in resonance, strongly coupled to the rate of damping, the moving area becomes smaller, while the maximum deflection becomes larger.

On theoretical grounds it is no mystery that this “wavy movement” of the BM is always running from the round window [base] towards the helicotrema [apex] of the cochlea.

It is a locally bound reaction behavior on a universally existing stimulus.
Using the material specifications this behavior can be calculated in a perfect way.

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

sequentially-scrolling-book-page044 sequentially-scrolling-book-page045 sequentially-scrolling-book-page046

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

figures figure 5

Applying Physics Makes Auditory Sense

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

fig-5 fig-5-txt

 

An animation movie of it:

An animation movie of it:

Windows Media (WMV) is a video

which-results-in-the-following-animation