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Author Topic: Can this relationship be derived between Schrodinger equation and Doppler shift?  (Read 12906 times)

Offline timey

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Can this relationship be derived between Schrödinger equation and Doppler shift equation?

Looking at Chladni plates and the resulting node patterns versus a laser beam trained on a mirror super glued to a speaker cone, causing the dot of the laser beam to form a pattern when a tone/frequency is applied - could there be a relationship between node patterns and lissajous curves?

Let's say we have a Chladni plate of a reflective/mirrored surface, and having established node patterns at varying frequencies, would lissajous curves (?)  be established at the same frequencies by pointing a laser at the surface?

And considering that the node pattern is caused at the points of least vibration of that frequency, and that the laser is depicting movement caused by the extremities of the vibration of that frequency - could it be said that the patterns created are the inverse of each other?
« Last Edit: 04/10/2016 13:16:38 by chris »


 

Offline evan_au

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could there be a relationship between [Chladni] node patterns and lissajous curves?
Chladni patterns are caused by the resonance of a physical object when subjected to a drive at a single frequency. This highlights the areas of high and low amplitude vibrations in the plate.

Lissajous figures are created when you drive a light by two different frequencies in the X and Y axes, where those frequencies are harmonically related to each other.

They can both produce interesting patterns, but they are different patterns.

Quote
Can this relationship be derived between Schrödinger equation and Doppler shift equation?
The Chladni patterns are derived from physical resonances of a macroscopic object with varying shape, thickness and (in the case of wooden instruments) the wood grain, glue, bracing struts, lacquer and sound holes. By varying the frequency of the incoming sound, you could graph the spectral response of resonances to the incoming sound.

Doppler shift of a laser beam can be used to measure the frequency of vibration of the object from which it is reflected. By graphing this in two dimensions, you could reconstruct the Chladni pattern.

Schrödinger's equation describes the electron energy levels of an atom. By exciting the atom with difference frequencies of light, you could graph the spectral response of resonances to the incoming light.

You may try (with great difficulty) to construct an instrument that has similar resonances to the Hydrogen spectrum. If the Hydrogen spectrum had energy levels related to 1/n, this may even be compatible with the Pythagorean ideal of harmony.

However, the actual energy levels of Hydrogen are related as 1/n2, so the harmony could be described at best as "complex", but most people would just call it "noise".
See: https://en.wikipedia.org/wiki/Rydberg_formula#Rydberg_formula_for_hydrogen
« Last Edit: 07/10/2016 21:39:39 by evan_au »
 

Offline timey

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Doppler shift of a laser beam can be used to measure the frequency of vibration of the object from which it is reflected. By graphing this in two dimensions, you could reconstruct the Chladni pattern.

It is this part of what you say that is of most interest to me, but first:

With regards to the Chladni plate node patterns...

You say that Lissijous curves or figures are caused by the harmonics created by driving 2 frequencies, but that the node patterns on the plate are caused by driving only 1 frequency...  Lets examine this a bit.

https://m.youtube.com/watch?v=lRFysSAxWxI

Here we see that there are 4 differently structured plates being used.  If you hang each plate as a gong and hit it with a gong striker, each plate will have its own naturally occurring frequency.  As far as I can tell, please correct me if wrong, what the bow is doing is adding another frequency to the plate that resonates with the plates natural frequency.  As the video shows, by bowing at different placements of the rim of the plate, the bow can add one of a few different frequencies which resonate/harmonise with the plates natural frequency.  But there are only so many frequencies that will harmonise with the plates own natural frequency.  Note how the placement of thumbnail on an exterior line of the node pattern, and the placement of the bow on, or between node lines changes the pitch in steps.

https://m.youtube.com/watch?v=wvJAgrUBF4w

Here we can see that this plate structure can only cause node patterns when subject to certain frequencies.  The frequencies that can achieve node patterns on this structure are surely mathematically linked to the structures own natural frequency?  Are all the frequencies that create node patterns on this structure following scale to the structures own frequency?

If so then the Chladni pattern is also created by driving 2 frequencies to harmonise/resonate, surely?

https://m.youtube.com/watch?v=RxzMzSZF_b4

Correct me if I'm wrong but these laser patterns must be caused by Doppler shift right?

https://m.youtube.com/watch?v=t6nGiBzGLD8

To get further understanding I watched the above.  Please note the use of the timing function.*

Correct me if I am wrong, but isn't the node pattern's association with Schrödinger due to standing wave function in that a wavelength can only fit x amount of times within a confine?

https://en.m.wikipedia.org/wiki/Wave_function_collapse

Quote:
"Significantly, the combined wave function of the system and environment continue to obey theSchrödinger equation.[4]"
Unquote:

* http://physics.stackexchange.com/questions/29551/quantum-explanation-of-doppler-effect

The second answer in relation to the first is interesting in relation to SR comments and time perturbations.

As you pointed out, node patterns are created in the areas of least vibration in the plate.
The Lissijous patterns are being created by the extremities of the vibration of the mirror.
I realise, as you have pointed out, that the patterns are not the same patterns, but given that each produced a pattern associated with the same frequencies, it interested me if one would be the inverse of the other?

So... If the Lissijous pattern is being caused by the Doppler shift of the laser beam, how would one mathematically graph this in two dimensions to reconstruct the Chladni pattern?
 

Offline Colin2B

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As far as I can tell, please correct me if wrong, what the bow is doing is adding another frequency to the plate that resonates with the plates natural frequency.  As the video shows, by bowing at different placements of the rim of the plate, the bow can add one of a few different frequencies which resonate/harmonise with the plates natural frequency. 
The bow doesn't add a frequency, it adds vibrational energy that allows the plate to resonate at one of its natural frequencies. Each frequency creates a different pattern caused by the standing waves. Lower order modes - those with fewer node lines - require less energy to initiate them, the simplest and lowest order being the cross/star pattern, which occurs at the lowest frequency. By bowing with differing amounts of force, ie energy, the bower can cause higher order modes to appear, you will also note that he uses his thumb nail to force a node at certain locations which produces a different mode and can bow at different points to encourage different antinodes.
What isn't clear from these videos is that the patterns are more complex than they appear at first sight. Although we see only antinodes, one antinodes might be going up while another goes down. Sometimes adjacent corners will move in opposite directions as the plate twists.
« Last Edit: 05/10/2016 00:04:24 by Colin2B »
 

Offline timey

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OK - I'm following your logic, and agree that there is evidence in the video that suggests that some of the complexity of how the patterns change from one to another is lost...

But Colin, isn't stating the bowing adding an energy to the situation in as much as saying the bow is adding a frequency?  Energy has a frequency, right?  Bowing with added energy, or at a placement on the edge of the plate that causes higher or lower energy, or by placing a thumbnail at a node point to alter the distribution of the added energy, these energies are all accompanied by their associated frequencies.  Note that some of the bowing ie: added energy/frequency does not cause node patterning.  The bowing has to be just so, or a tone/node pattern cannot be achieved.

In the second video it is shown that frequencies are being driven to the plate directly.  This can also be considered to be a case of adding vibrational energy, but the machine kindly translates this added energy into its associated frequency for us.  Note that the node patterns are only formed fully at specific frequencies, albeit some of the complexity as the patterns change seems to be lost.

My thoughts drifting towards thinking (subject to being wrong of course) that these specific frequencies that the patterns are fully formed at are harmonics to the plates own frequency.  A frequency that is related to the plates own energy of e=mc^2.

So in light of what I've mentioned, do you think a Chladni plate pattern is involving 2 frequencies, like the Lassijous figures do, or is it just 1 frequency?
 

Offline evan_au

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Quote from: timey
do you think a Chladni plate pattern is involving 2 frequencies, like the Lassijous figures do, or is it just 1 frequency?
I was thinking of exciting the vibrations with a sine-wave generator, which is just 1 frequency.

If you were adding energy with a violin bow, I guess you could imagine a rectangular plate with sides in a 3:1 ratio.
If you bowed near a corner, I guess you could set up vibrations which are harmonically related in different axes, and use that to produce Lissajous figures.

Even when you are bowing a violin string, it is normal to produce several harmonics in the same string, simultaneously.

If you activated the Chladni pattern with a loudspeaker, you could feed in two different tones to excite two different resonances at the same time.

Quote
isn't stating the bowing adding an energy to the situation in as much as saying the bow is adding a frequency?
Colin is the luthier here, but my simple understanding is that the keratin flakes of the horse-hair have a sawtooth edge, which repeatedly pulls and releases the string (or plate, in this case). This represents a fairly broadband stimulation of any resonances.

These frequent small knocks can build up into a large oscillation at the natural resonant frequency of the object.
- I assume the location, pressure, speed and direction of the bowing can accentuate or inhibit particular vibration frequencies.
- Or it can produce a chaotic screech, like most beginner violin players...
 

Offline Colin2B

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My thoughts drifting towards thinking (subject to being wrong of course) that these specific frequencies that the patterns are fully formed at are harmonics to the plates own frequency.  A frequency that is related to the plates own energy of e=mc^2.
….
So in light of what I've mentioned, do you think a Chladni plate pattern is involving 2 frequencies, like the Lassijous figures do, or is it just 1 frequency?
You're in the right direction but veering off to left field.
I may have assumed understanding of what vibrational modes involve, so perhaps we need to step back a bit.
Let's take the standard vibrating string in every school textbook. Fixed at both ends if we pluck the middle we get that characteristic shape of antinode in the middle and nodes at each end. You might be excused for thinking that the centre of the string is just moving up and down, but it's more complex than that. As Evan says, all vibrations of this type involve more than one frequency, in this case when plucked at the centre the string will vibrate in all modes where there is an antinode at the centre and nodes at each end and if you draw them out you will find they are the odd harmonics of the string fundamental. In theory you should have all the odd frequencies, but as fourier analysis shows the components of a square wave are the odd harmonics up to infinity and the mechanics of the string mean that it can't support all the harmonics. If you superpose the harmonics up to about 9 you will get an envelope very similar to the one in the textbooks showing the average excursion of the points along the string – average excursion because the harmonics get out of phase so the string ends up doing some interesting contortions rather than the implied moving up and down at the centre.
Plucking at different points eg ¼ along the string, will excite a different series of harmonics.

When you look at analysing the Chladni plates you can view them as 2 dimensional open ended pipes – antinodes at the outer edge. As with strings, vibrating pipes and plates all generate multiple frequencies, but it is the mix of frequencies that give the specific patterns on the plates.

isn't stating the bowing adding an energy to the situation in as much as saying the bow is adding a frequency? 
As you can see from the string example, plucking doesn't add a frequency if just allows the string to vibrate in its natural manner.
Let's look at a simple plate, a school ruler. If you hold one end on the desk and twang the end the ruler will vibrate at its natural frequency. If you bow the end of the ruler (ie plate) the rosin on the bow grips the end of the ruler and pushes it down, when the return force of the spring in the ruler overcomes the grip of the bow the ruler springs back and again will vibrate at its natural frequency.
The bow isn't adding a frequency, you could just as easily twang the edge of the Chladni plate or tap it with a stick, all would cause it to vibrate in a particular mode depending on where you hit it. However, you would have to twang it a lot of times before the sand moved to the nodes, the bow or a loudspeaker is easier.

E=mc2? No. There is mass of the plate, which affects the frequency of the modes. There is a speed which is the wave speed of sound across the plate, that is largely determined by E, Young's Modulus. But we aren't dealing here with mass energy equivalence, much more basic.

Schrodinger? Well, the mathematical tools used in acoustic analysis eg fourier analysis, eigenvalues etc are all part of QM, but the Shrodinger  equation is a very specific wave analysis not part of the analysis of string and plate vibrations.

 

Offline timey

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Colin - although your description of harmonics is highly relevant, it is something that will be obvious to anyone who has played a string instrument (Yup, I was a screechy violinist as a child Evan...(chuckle)) ...or inspected the inner workings of a piano, which I do and have.
(although I must say, in that I have worked with music, that I was literally horrified when watching the oscilloscope video at the extent of that which I didn't know and probably still don't know...)
No instrument makes a sound of itself.  An instrument always requires an input of vibration to resonate with.  All vibrations have an associated frequency, and all a frequency is, is the amount of vibrations per standard second.

Schrodinger? Well, the mathematical tools used in acoustic analysis eg fourier analysis, eigenvalues etc are all part of QM, but the Shrodinger  equation is a very specific wave analysis not part of the analysis of string and plate vibrations.

What you are saying here is not really correct.  Please see link:

https://en.m.wikipedia.org/wiki/Ernst_Chladni

Quote:
" In quantum mechanics, Chladni figures ("nodal patterns") are known to be related to the solutions of theSchrödinger equation for one-electron atoms, and the mathematics describing them was used by Erwin Schrödinger to arrive at the understanding of electron orbitals.[12]"
Unquote:

If this venture into advanced mathematics is being off left side then off left side is exactly where I want to be...

A Chlani plate node pattern relates to the Schrödinger equation - and the Lissijous figures, by the nature of the mechanics that are involved in their creation, must relate to the Doppler shift equation.

A node pattern is caused at the areas of least vibration.
A Lissijous figure is caused by the areas of most vibration.
The question being: Could node patterns be the opposite, or inverse, of Lissijous figures?

I am looking for the mathematical relationship between these patterns and am thinking that "phase shift' might be relevant?
 

Offline jeffreyH

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Colin - although your description of harmonics is highly relevant, it is something that will be obvious to anyone who has played a string instrument (Yup, I was a screechy violinist as a child Evan...(chuckle)) ...or inspected the inner workings of a piano, which I do and have.
(although I must say, in that I have worked with music, that I was literally horrified when watching the oscilloscope video at the extent of that which I didn't know and probably still don't know...)
No instrument makes a sound of itself.  An instrument always requires an input of vibration to resonate with.  All vibrations have an associated frequency, and all a frequency is, is the amount of vibrations per standard second.

Schrodinger? Well, the mathematical tools used in acoustic analysis eg fourier analysis, eigenvalues etc are all part of QM, but the Shrodinger  equation is a very specific wave analysis not part of the analysis of string and plate vibrations.

What you are saying here is not really correct.  Please see link:

https://en.m.wikipedia.org/wiki/Ernst_Chladni

Quote:
" In quantum mechanics, Chladni figures ("nodal patterns") are known to be related to the solutions of theSchrödinger equation for one-electron atoms, and the mathematics describing them was used by Erwin Schrödinger to arrive at the understanding of electron orbitals.[12]"
Unquote:

If this venture into advanced mathematics is being off left side then off left side is exactly where I want to be...

A Chlani plate node pattern relates to the Schrödinger equation - and the Lissijous figures, by the nature of the mechanics that are involved in their creation, must relate to the Doppler shift equation.

A node pattern is caused at the areas of least vibration.
A Lissijous figure is caused by the areas of most vibration.
The question being: Could node patterns be the opposite, or inverse, of Lissijous figures?

I am looking for the mathematical relationship between these patterns and am thinking that "phase shift' might be relevant?

Well done.
 

Offline alancalverd

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A Chlani plate node pattern relates to the Schrödinger equation - and the Lissijous figures, by the nature of the mechanics that are involved in their creation, must relate to the Doppler shift equation.

A node pattern is caused at the areas of least vibration.
A Lissijous figure is caused by the areas of most vibration.
The question being: Could node patterns be the opposite, or inverse, of Lissijous figures?

Lots of muddled thinking here.

Chladni figures are the inverse distribution of amplitude over a resonating body, and can in some cases resemble atomic orbitals. 

Lissajous figures are the patterns produced by the intersection of two sinusoidal curves the axes of which are at right angles to each other. Where the frequencies are in a harmonic relationship such that the phases of the two curves are locked, the figure appears static. Nothing to to with resonance or orbitals, just the 2D solution of two simultaneous equations. And although not truly Lissajous, you can use a similar XY plot on an oscilloscope to investigate the phase relationship of any two periodic signals.

The Doppler shift equation is a scalar only. df/f = v/c where df is the change in observed frequency, v is the relative velocity of source and observer, and c is the velocity of sound/light/whatever. No sinusoids or spatial coordinates.   
 

Offline Colin2B

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What you are saying here is not really correct.  Please see link:

https://en.m.wikipedia.org/wiki/Ernst_Chladni

Quote:
" In quantum mechanics, Chladni figures ("nodal patterns") are known to be related to the solutions of theSchrödinger equation for one-electron atoms, and the mathematics describing them was used by Erwin Schrödinger to arrive at the understanding of electron orbitals.[12]"
Unquote:

Yes, I know the history which I why I made the comment about the commonality of the maths tools.  However, as I said the Shrodinger equation has a very specific application and you can't turn it round and use it to analyse Chladni patterns or reach conclusions about Doppler shift on the laser beam. As Evan pointed out:
Schrödinger's equation describes the electron energy levels of an atom. By exciting the atom with difference frequencies of light, you could graph the spectral response of resonances to the incoming light.
You may try (with great difficulty) to construct an instrument that has similar resonances to the Hydrogen spectrum. If the Hydrogen spectrum had energy levels related to 1/n, this may even be compatible with the Pythagorean ideal of harmony.
However, the actual energy levels of Hydrogen are related as 1/n2, so the harmony could be described at best as "complex", but most people would just call it "noise".

Returning to your original question:
..considering that the node pattern is caused at the points of least vibration of that frequency, and that the laser is depicting movement caused by the extremities of the vibration of that frequency - could it be said that the patterns created are the inverse of each other?
Pointing a single laser at the mirror surface won't create an inverse of the pattern as you need a way of mapping the areas of maximum displacement (as the inverse of the map of nodes), however you could use laser holography to do it.

PS We'll have to agree to disagree on whether the bow is inputting a frequency or just activating the natural resonance modes of the plate.



 

Offline timey

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Quote
A Chlani plate node pattern relates to the Schrödinger equation - and the Lissijous figures, by the nature of the mechanics that are involved in their creation, must relate to the Doppler shift equation.

A node pattern is caused at the areas of least vibration.
A Lissijous figure is caused by the areas of most vibration.
The question being: Could node patterns be the opposite, or inverse, of Lissijous figures?

Lots of muddled thinking here.

Chladni figures are the inverse distribution of amplitude over a resonating body, and can in some cases resemble atomic orbitals. 

Lissajous figures are the patterns produced by the intersection of two sinusoidal curves the axes of which are at right angles to each other. Where the frequencies are in a harmonic relationship such that the phases of the two curves are locked, the figure appears static. Nothing to to with resonance or orbitals, just the 2D solution of two simultaneous equations. And although not truly Lissajous, you can use a similar XY plot on an oscilloscope to investigate the phase relationship of any two periodic signals.

The Doppler shift equation is a scalar only. df/f = v/c where df is the change in observed frequency, v is the relative velocity of source and observer, and c is the velocity of sound/light/whatever. No sinusoids or spatial coordinates.   

Sorry, no muddled thinking here on my part.  You just have not bothered to read the thread and therefore are unaware of the context.

Please read post 2 and observe the videos that have already outlined what you have posted, and the link to physics stack exchange and answer 2 in relation to answer 1, which will give you the context.  I am thinking that phase shift is the common ground.

There are some outstanding issues that are a bit muddling though and perhaps you can indeed help.
The first being:
-is a Chladni plate being driven to resonance by 2 frequencies? ie:  its own inherent frequency/tone resonating with the frequency associated with the input energy... to create harmonics of the plates natural frequency as the input  frequency is increased.  Or is there just 1 frequency involved in a Chladni pattern, as Colin says?
 

Offline alancalverd

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Back to the "frequency" of a chladni plate. It doesn't have an inherent  single frequency. All that is required of a vibrating plate is that there is a node(s) at the point(s) of suspension. You can then, in principle, set up any number of standing waves whose nodes correspond, by bashing it hard enough. Continuous input to a symmetrcal plate by a violin bow will select for those harmonics with antinodes at the point of contact. If you are very good at bowing, you can select just the fundamental of a circular plate, which will give you a circular pattern with a single frequency only. 
 

Offline timey

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When the plate is not being vibrating, (ie: does not have an input) the structure already contains standing waves...doesn't it?
 

Offline Colin2B

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When the plate is not being vibrating, (ie: does not have an input) the structure already contains standing waves...doesn't it?
No, there is no standing wave until the plate is vibrating at the specific resonant frequency for that standing wave(s).
« Last Edit: 06/10/2016 18:53:41 by Colin2B »
 

Offline timey

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So the plate does not have an energy associated with its mass, and De Broglie matter waves are only mathematical conveniences with which to calculate quantum probabilities?

Or do you mean that there is no visible evidence of standing waves in the structure until an input frequency resonates with one or more of the inherent standing waves harmonics...

Otherwise what in the plate is the input resonating with please?
 

Offline Colin2B

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So the plate does not have an energy associated with its mass, and De Broglie matter waves are only mathematical conveniences with which to calculate quantum probabilities?
I don't see the relevance of this. We don't need to use quantum level calculations to calculate vibrations in large scale structures. Of course the plate has an energy associated with its mass, but we are not looking to convert any of that mass to energy or vise versa, neither are we interested in its relativistic kinetic energy.

Or do you mean that there is no visible evidence of standing waves in the structure until an input frequency resonates with one or more of the inherent standing waves harmonics...
No, I mean that there are no standing waves until the plate vibrates. Standing waves are vibrations, they are the characteristic patterns of the resonances.
The input frequency does not resonate with anything, it is the plate which resonates at certain frequencies. The input frequency doesn't change its form.

Otherwise what in the plate is the input resonating with please?
I think there is a confusion of terminology here and understanding of resonance so it might help to step backwards and forget about input frequencies for a moment.

Imagine you ping the edge of the plate with a plectrum. The plectrum pushes down the edge and releases it, the speed at which the edge returns towards its undeformed position will determine the frequency at which the plate 'rings', this speed is in turn determined by the stiffness of the plate (its springyness) and its mass (inertia) and the dimensions of the plate. This is an oversimplification, but it is intended to illustrate that there is nothing inside the plate which resonates, but it is the plate itself - governed by its physical characteristics - which resonates. Also, it does not need an input frequency in order to resonate just some action to set it going.
The use of a loudspeaker as an initiator is useful because it provides a continuous stimulus and also allows us to see at which frequencies the resonances occur.
 

Offline timey

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I understand your take in the sense of analysing sound Colin, but what I'm interested in is the relationship between the Chladni patterns and the Schrödinger equations, the related analysis of wave function at quantum level, the relationship that the Lorentz transformations have in relation to Doppler shift at quantum level, the time perturbations that are used to calculate at quantum level...
And the relationship that Lissajous figures have with the electron.

Any energy added at this level is significant to frequency and therefore is relevant to standing wave function.  The properties of the plate are significant because they are quite simply properties that are inherent to the plate and these properties affect the effects of any input energy. (Energy mass equivalence asside for time being)

Here we can see that Lissajous figures can also be traced out by sand dripping from a compound pendulum.

https://www.britannica.com/topic/Lissajous-figure

http://www.differencebetween.com/difference-between-simple-pendulum-and-vs-compound-pendulum/

Quote:
" What is the difference between Simple and Compound Pendulums?
• The period and, therefore, the frequency of the simple pendulum depends only on the length of the string and the gravitational acceleration. The period and the frequency of the compound pendulum depend on the length of gyration, the moment of inertia, and the mass of the pendulum, as well as the gravitational acceleration.
• The physical pendulum is the real life scenario of the simple pendulum."
Unquote:

An apparatus of frequency vibrated mirrors causing a laser beam to trace out Lissajous figures is physically equal to the actions of a compound pendulum...
Lets say the mirror was engineered to be resonant at a pure tone (the same tone as was used to create the Lissajous figure) and sand were placed on the mirror being vibrated (at same other tone that created the Lissajous figure) ...then a Chladni pattern would emerge...?
And if so, what would the mathematical relationship be?
« Last Edit: 07/10/2016 10:01:21 by timey »
 

Offline Colin2B

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...it is the relationship between the Chladni patterns and the Schrödinger equations, the related analysis of wave function at quantum level, the relationship that the Lorentz transformations have in relation to Doppler shift at quantum level, the time perturbations that are used to calculate at quantum level...
And the relationship that Lissajous figures have with the electron.
Ok, I may have misunderstood what you are trying to do here. It sound like you are trying to draw some conclusions about actions at the atomic level using chladni, lissajou and Doppler, rather than trying to use shrodinger, e=mc2, etc to analyse the plates.

Here we can see that Lissajous figures can also be traced out by sand dripping from a compound pendulum.
This type of compound pendulum is in fact 2 simple pendulums linked, one oscillates as a sine wave in x direction the other in y so that their combined motion is the lissajou curve. Sorry to labour this but it is important to recognise that the curves are the result of combining 2 pure sinewaves.

An apparatus of frequency vibrated mirrors causing a laser beam to trace out Lissajous figures is physically equal to the actions of a compound pendulum...
Yes, you could cause the mirror to vibrate on 2 axes perpendicular to each other across the face of the mirror, each axis would need to vibrate with a sinewave.

Lets say the mirror was engineered to be resonant at a pure tone (the same tone as was used to create the Lissajous figure) and sand were placed on the mirror being vibrated (at same other tone that created the Lissajous figure) ...then a Chladni pattern would emerge...?
And if so, what would the mathematical relationship be?
To create a lissajou figure you need 2 sinewaves not one.
The plates do resonate at more than one frequency and it is possible to make them resonate at 2 frequencies simultaneously by feeding 2 tones into a loudspeaker.
If you take the example of the basic mode of a square plate, free at the edges (I think you used this eg somewhere) at one frequency the pattern is a cross with nodes at the centre of each face, at a second frequency the pattern is a cross with the nodes at the corners of the plate. If you force the plate to vibrate at both of these frequencies simultaneously you will get a combination of the 2 patterns i.e. 2 superimposed stars.
You could also try building a plate in the same proportion as the lissajou frequencies so that the resonating length/width is the same as the ratio of the sinewaves e.g. 2:1. In this case the rectangular plate pattern is a simple 2 nodes across the width of the plate dividing it into 3 antinodes - the ends go up and the centre goes down etc, this is the same pattern as a rectangular bar. Also this pattern doesn't change with the ratio of width to length unlike the lissajou figure.
The fact is that the lissajou curves and the chladni patterns are 2 different phenomena, giving us different information.
If you want to try applying lissajou curves to qm perhaps you ought to look for related frequencies, say, 2 harmonic oscillators to see whether they exhibit something similar to the curves.
 

Offline timey

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Yes Colin - you are on the right track, but let me more fully explain my direction...

The Chladni patterns are already connected to the Schrödinger equation.  Schrödinger used the mathematics of Chladni patterns to work out the atomic orbital of the electron.

The Schrödinger equation is related to the Doppler shift equation via the Lorentz transformations, and perturbations of time are used to calculate the probability of quantum actions.

The common ground in these instances is phase shifting of either space or time.

In that Lissajous patterns can be created by reflected light being vibrating off a mirror, and this is almost synonymous to the Pound Rebka gamma ray emitter being vibrated by frequency tones creating a Doppler shift in the test signal of the experiment - Lissajous figures must have some relationship with the Doppler shift equation and therefore a connection to the gravitational shift equation.

This corresponds with the compound pendulums connection with Lissajous figures because gravity is one of the input sine waves of the compound pendulum.

Lissajous figures also relate to the electron via memoristor and are 'already' connected to quantum.

*

Yes - you are entirely correct...  I won't actually be doing this myself because I haven't got a pot to p*ss in and I'd sooner see a dinner on my plate, but yes it would be possible to create a plate that, much like xylophone notes, are engineered to resonate at a pure tone... Inputting that plate with another frequency that is known to create a Lissajous figure when combined with this pure tone, the Chladni pattern that emerges on the plate will not be the same pattern as a the Lissajous figure, but both patterns, in that they are corresponding to the use of the same frequencies, will have a mathematical relationship - and that relationship would be very interesting.

By the physical nature of how the patterns are created, the Lissajous patterns being the result of light thrown outwards into patterns from the extremities of the vibrations, and the Chladni patterns being the result of sand being pushed into the areas of least vibration, the Chladni pattern should, by all logic, be an inverse representation of the corresponding Lissajous figure.
 

Offline jeffreyH

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This is the wrong question anyway. How does negative kinetic energy relate general relativity to quantum mechanics?
 

Offline timey

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Perhaps, but while not detracting from Dirac's contribution, on the basis that the knee bone is connected to the thigh bone, the question could also be a lot of things, the most direct form of which being: Can quantum and the standard model be united with gravity?

I notice that nobody has commented to confirm or deny my observations on the difference between the physical mechanics creating these patterns...

Never mind or not the notion of 'how' a Chladni plate pattern is created, just 'what' the pattern is and the mathematics of the points of least vibration, I would have thought that it wouldn't be that much of a mathematical venture to analyse Lissajous patterns as to the points of least vibration on the basis that the pattern is being created from the points of most vibration?
 

Offline alancalverd

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The coincidence of mathematical models does not necessarily indicate any fundamental connection between the phenomena they model. Tide tables look like (indeed are) multiple superpositions of sine waves and the Southampton tide in particular looks like a simple Lissajous figure as the primary Atlantic tide is modulated by the Isle of Wight, but you'd have a hell of a job predicting it from the shape of hydrogen bonds in seawater.

The use of a sine wave drive in the Pound-Rebka experiment has nothing to do with resonance or interference in the loudspeaker cone - almost the opposite. If you apply a drive current I = a sin(ft) where a is the amplitude and f is a frequency within the linear single mode response of the cone (i.e a low frequency), the velocity of the cone at any time t is a cos (ft). If you use a phase-sensitive detector to correlate the received signal with I you can find the value of t that produces maximum signal and hence calculate the critical Doppler velocity.

The universe is made up of zillions of things (some of which we have never observed) all interacting with each other in umpteen ways (some of which we don't understand). Physics equations rarely invoke more than sine waves and x^2, so they tend to be a bit repetitive: representing the entire orchestra, choir and cathedral acoustics with one bar of dots doesn't equate Lloyd Webber with Haydn.   

When you do succeed, I will of course be delighted to have that sentence quoted in your Nobel speech, but if I wanted to produce a predictive model of quantum gravity, I wouldn't start with Doppler or Chladni.
« Last Edit: 10/10/2016 13:08:17 by alancalverd »
 

Offline Colin2B

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I notice that nobody has commented to confirm or deny my observations on the difference between the physical mechanics creating these patterns...
Sorry, been very busy for other than short answers. Will respond when I get a moment.
Need to continue in new theories as your posts have moved a long way over there.

Edit - aAlan responded while I was typing, but my thoughts are similar. I'll try to go through later.
« Last Edit: 10/10/2016 08:27:50 by Colin2B »
 

Offline timey

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I notice that nobody has commented to confirm or deny my observations on the difference between the physical mechanics creating these patterns...
Sorry, been very busy for other than short answers. Will respond when I get a moment.
Need to continue in new theories as your posts have moved a long way over there.

Edit - aAlan responded while I was typing, but my thoughts are similar. I'll try to go through later.
Hey Colin - no need for apology.

Actually the question "Can quantum and the standard model be united with gravity?" ...is older than I am, but New Theories is OK with me.

http://www.physicsclassroom.com/class/sound/Lesson-4/Standing-Wave-Patterns

It clearly states in this link that all objects resonate with their own natural frequency or frequencies.  That it is driving these natural frequencies with an outside source that causes standing waves.  That the patterns are only created within the medium at specific frequencies of vibration.  And these specific frequencies are harmonics of the natural frequencies of the object.

Therefore since 1 frequency is being driven with another frequency, there 'must' be 2 frequencies involved in creating the Chladni pattern.

Is there anybody reading who can agree on this point?
 

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