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Author Topic: Can this relationship be derived between Schrodinger equation and Doppler shift?  (Read 12873 times)

Offline timey

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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.

Alan - It is a text book fact that Schrödinger used the mathematics of Chladni patterns to work out the atomic orbital of the electron.  It is also a text book fact that Lissajous patterns have a connection to the electron.  Asking questions of coincidences in physics such as this is not irrelevant, although clearly questions asked may turn out to be irrelevant.  However, without asking the questions one would not know the answers, and even questions being found to have no relevance has relevance in itself.

You said:
"The use of a sine wave drive in the Pound-Rebka experiment has nothing to do with resonance or interference in the loudspeaker cone"

Correct.  Its the actions of the speaker cone vibrating the gamma ray emitter (Pound Rebka), the laser dot on mirror (Lissajous figures),  that cause Doppler shift.

I'm looking at the difference in these Chladni and Lissajous patterns as being due to opposite phase shifting.

Opposite phase shifting becomes relevant to what I am saying with my theory of time and the fact that physics measures physical actions via the standard second, and my theory states the t of any mass situation as being t is equal to the phase period of its f, and any space situation as being t is equal to the phase period of the hypothetical f of the energy of the gravity field at distance from mass.

Much to my disappointment though Alan, my experience with you has led me to the understanding that if it's not already written in a text book, or is not something that yourself and Kibblewhite discussed, then it would seem to be not something that you are capable of considering without prejudice.

So Chladni patterns and Doppler shift is not where you would start in a quest to unite quantum with gravity...  Just for the record, where exactly did "you" start from then?
 

Online alancalverd

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There is no essential connection between Lissajous and Doppler. Your connection between Lissajous and the electron would be interesting to discover. Whatever you do in science, don't confuse the mathematical model with the reality - especially if the model is continuous and the reality is quantised.
 

Offline Colin2B

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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.
That's the point ive been trying to make all along. It is the sum of these harmonics that causes the standing waves. You don;t need a second frequency, and the outside source doesnt need a frequency component - think blowing across the top of a bottle, yiu are only blowing not whistling or huming a frequency. You may have missed what I was saying or Im not explaining clearly enough.
 

Offline timey

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There is no essential connection between Lissajous and Doppler. Your connection between Lissajous and the electron would be interesting to discover. Whatever you do in science, don't confuse the mathematical model with the reality - especially if the model is continuous and the reality is quantised.
Light bouncing off a mirror being vibrated causes a Doppler shift in the light.

Lissajous figures are connected to the electron via memorista.

Whatever you do in science Alan don't confuse a mathematical model with reality, especially if the model is quantised and reality is continuous!
 

Offline timey

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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.
That's the point ive been trying to make all along. It is the sum of these harmonics that causes the standing waves. You don;t need a second frequency, and the outside source doesnt need a frequency component - think blowing across the top of a bottle, yiu are only blowing not whistling or huming a frequency. You may have missed what I was saying or Im not explaining clearly enough.
No Colin - you explain yourself well and I have missed nothing...
You however seem to be missing the fact that an object has an associated energy, or energies, and that energy has an associated frequency.  And that any energy added to an object in any fashion has an associated frequency.  Some methods of adding energy such as being hit singularly with a rubber mallet will only change amplitude if more energy is given to the force of the strike, but adding energy that consists of an increasing frequency will cause changes in standing waves confined within the object.

Any change in an objects behaviour due to an energy input is an interaction between the energy of the input, and the energy or energies of the object.

Therefore there is the frequency or frequencies of the object, and the frequency of the input signal at play in the creation of Chladni patterns, surely...

How can one logically deduce otherwise?
 

Online alancalverd

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Assuming you mean memristor, I'd appreciate a reference to how they connect electrons to Lissajous figures. Curious devices, they seem to have disappeared from sight in the last 40 years.

Quote
You however seem to be missing the fact that an object has an associated energy, or energies, and that energy has an associated frequency.
Alas that is not true or even generally meaningful.

And, sadly, the Lissajous figures created by reflecting light from a vibrating mirror have nothing to do with Doppler shift. It's just the path of the reflected beam when the mirror rotates about two axes.   

 
« Last Edit: 10/10/2016 22:44:14 by alancalverd »
 

Offline jeffreyH

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Offline timey

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Assuming you mean memristor, I'd appreciate a reference to how they connect electrons to Lissajous figures. Curious devices, they seem to have disappeared from sight in the last 40 years.

Quote
You however seem to be missing the fact that an object has an associated energy, or energies, and that energy has an associated frequency.
Alas that is not true or even generally meaningful.

And, sadly, the Lissajous figures created by reflecting light from a vibrating mirror have nothing to do with Doppler shift. It's just the path of the reflected beam when the mirror rotates about two axes.   

https://en.wikipedia.org/wiki/Lissajous_curve

quote:
"A Lissajous curve is used in experimental tests to determine if a device may be properly categorized as a memristor."

https://en.wikipedia.org/wiki/Memristor#Experimental_tests_for_memristors

please see: experimental tests for memristor

Quote:
"Chua and his student Sung Mo Kang generalized the theory of memristors and memristive systems including a property of zero crossing in the Lissajous curve characterizing current vs. voltage behavior."

https://learn.sparkfun.com/tutorials/voltage-current-resistance-and-ohms-law

"The three basic principles for this tutorial can be explained using electrons, or more specifically, the charge they create: Voltage is the difference in charge between two points. Current is the rate at which charge is flowing. Resistance is a material's tendency to resist the flow of charge (current)"

https://prezi.com/k67jmml5iopb/applications-of-lissajous-figures/

I did not watch this as it will not play on my phone, but the write up says: 
"with very difficult math approach like Hamiltonian in Quantum mechanics"

*

All mass is made of atoms, which contain electrons, which have energy.

*

Ah, ok...good, good!  Not a Doppler shift.  Lets examine that.  The Pound Rebka attaches a gamma ray to a speaker cone and a Doppler shift is created in the test signal of the experiment.  If we attach a laser to the speaker cone then a Doppler shift would be created in the light of the laser.  So by pointing a laser at a mirror attached to a speaker cone, and pointing a laser 'straight' at it the light shining back would be Doppler shifted...

...But the Lissajous pattern is not created like this.  Looking at the tuning fork version, the laser beam  is pointed at a mirror set up at an angle that reflects the beam to another tuning fork with a mirror on it, that then reflects the beam onto the screen.  Both tuning forks are vibrating.  Both mirrors are adding a Doppler shift to the beam of laser light.

« Last Edit: 11/10/2016 01:33:42 by timey »
 

Offline timey

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There may be a confusion with phonons.

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

Interesting.  I'll be reading that closely.
 

Offline jeffreyH

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And try going to this web site and watch the video to see how the Chladni patterns relate to quantum mechanics. http://oyc.yale.edu/chemistry/chem-125a/lecture-9
 

Offline Colin2B

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There may be a confusion with phonons.

I agree Jeff, there is a big difference between the propagation of sound across a crystal lattice and the response of that lattice to mechanical distortion as in the Chladni plates. If we want to estimate the speed of sound in a material, we don't need to do an analysis of the atomic structure of the crystal lattice, we can use Youngs Modulus. I think there is also some confusion with phenomena like black body radiation:
You however seem to be missing the fact that an object has an associated energy, or energies, and that energy has an associated frequency.  And that any energy added to an object in any fashion has an associated frequency. 
There is no direct relationship between increasing the energy input and the resonant frequencies, in fact the resonances require less energy for a particular amplitude of vibration compared to none resonant frequencies. This is similar to the energy wells found in quantum models and that electrons find it easier to remain in certain orbitals.

adding energy that consists of an increasing frequency will cause changes in standing waves confined within the object.
In its quiescent state there are no flexing movements in the plate so no standing waves 'confined' in the plate.

Therefore there is the frequency or frequencies of the object, and the frequency of the input signal at play in the creation of Chladni patterns, surely...

How can one logically deduce otherwise?
As I explained earlier, the input frequency acts in the same way as hitting with a hammer, causing  a mechanical displacement, the returning force due to stiffness and the mass of the plate determines the resonance, hence the harmonics, hence the patterns. This is all due to the flexing of the plate and there is no underlying frequency which the input signal combines with.

This corresponds with the compound pendulums connection with Lissajous figures because gravity is one of the input sine waves of the compound pendulum.
No, there is no gravity sinewave. Gravity acts as the returning force, in the same way as the stiffness of the plate.

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.
As Alan said, the lissajous patterns are the locus of the sum of 2 sine waves, as such they will be the resultant of nodes and antinodes of the individual waves.

If you are going to draw parallels between mechanical phenomena an atomic/quantum behaviour you need to start with a correct model of the mechanical system.
« Last Edit: 11/10/2016 10:08:40 by Colin2B »
 

Offline evan_au

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As an aside...
Quote from: alancalverd
memristor.... Curious devices, they seem to have disappeared from sight in the last 40 years.
Research is still continuing - the latest fads seem to be in emulating neurons, or displacing flash drives...
See: http://spectrum.ieee.org/the-human-os/semiconductors/memory/mimicking-the-synapses-of-the-brain
 
The following users thanked this post: timey

Offline timey

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There may be a confusion with phonons.

I agree Jeff, there is a big difference between the propagation of sound across a crystal lattice and the response of that lattice to mechanical distortion as in the Chladni plates. If we want to estimate the speed of sound in a material, we don't need to do an analysis of the atomic structure of the crystal lattice, we can use Youngs Modulus. I think there is also some confusion with phenomena like black body radiation:

But I am not discussing the propagation of sound - I am discussing the distribution of resonant vibration.

Phonons are the product of perturbation theory as are all quantum calculations.  I'm not sure what confusion you refer to regarding the black body radiation phenomenon, but fact of the matter is that if one attempted to apply these time perturbations to the black body radiation data directly, then the quantised nature of the data would be eliminated at source.

You however seem to be missing the fact that an object has an associated energy, or energies, and that energy has an associated frequency.  And that any energy added to an object in any fashion has an associated frequency. 
There is no direct relationship between increasing the energy input and the resonant frequencies, in fact the resonances require less energy for a particular amplitude of vibration compared to none resonant frequencies. This is similar to the energy wells found in quantum models and that electrons find it easier to remain in certain orbitals.
adding energy that consists of an increasing frequency will cause changes in standing waves confined within the object.
In its quiescent state there are no flexing movements in the plate so no standing waves 'confined' in the plate.

The standing waves that are inherent in the plate are holding it in its non-vibrating position.  Without them the plate would not be a plate.

Therefore there is the frequency or frequencies of the object, and the frequency of the input signal at play in the creation of Chladni patterns, surely...

How can one logically deduce otherwise?
As I explained earlier, the input frequency acts in the same way as hitting with a hammer, causing  a mechanical displacement, the returning force due to stiffness and the mass of the plate determines the resonance, hence the harmonics, hence the patterns. This is all due to the flexing of the plate and there is no underlying frequency which the input signal combines with.

Well actually Colin, you have explained this scenario in both ways.  You have told me that a plate has natural frequencies, and the input to the plate has no frequency, in that one blows rather than whistles across the top of a bottle to create a sound, or hits with a hammer,(bowing with a violin bow is actually much more complicated physics)....
And then you tell me, as we can quite clearly observe in the links provided, that Chladni patterns are forming on a plate and changing from one pattern to another as an input is being increased in its frequency.

How the different tones on steel drums are constructed and how the standing wave of this tone resonating is confined to the tone zone is interesting Colin.   

This corresponds with the compound pendulums connection with Lissajous figures because gravity is one of the input sine waves of the compound pendulum.
No, there is no gravity sinewave. Gravity acts as the returning force, in the same way as the stiffness of the plate.

Gravity is one of the driving forces of the compound pendulum's representation of a Lissajous figure.

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.
As Alan said, the lissajous patterns are the locus of the sum of 2 sine waves, as such they will be the resultant of nodes and antinodes of the individual waves.

Precisely the connection that I am making with Chladni plates.
 
If you are going to draw parallels between mechanical phenomena an atomic/quantum behaviour you need to start with a correct model of the mechanical system.

My post to Alan concerning the tuning fork version of producing Lissajous patterns 'is' mechanically correct.
« Last Edit: 11/10/2016 12:47:54 by timey »
 

Offline timey

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Jeff - thanks for the links...

Please see first part of post to Colin above.

Please also see the 2014 link of applications of Lissajous figures in my last post to Alan, and the notion of maths approaching the Hamiltonian.

https://en.wikipedia.org/wiki/Quantum_gravity

quote:
" The Problem of Time
Main article: Problem of Time
A conceptual difficulty in combining quantum mechanics with general relativity arises from the contrasting role of time within these two frameworks. In quantum theories time acts as an independent background through which states evolve, with the Hamiltonian operator acting as the generator of infinitesmal translations of quantum states through time.[29] In contrast, general relativity treats time as a dynamical variable which interacts directly with matter and moreover requires the Hamiltonian constraint to vanish,[30] removing any possibility of employing a notion of time similar to that in quantum theory."
unquote:

Please note that Special Relativity also uses an independent time background.

Please note that the Shrodinger and Doppler shift are united in quantum via the Lorentz transformations, which phase both time and space.
 

Offline jeffreyH

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Then what is required, as I have stated in new theories, is a concept of action dilation rather than just time dilation. Then time is no longer an independent background in quantum mechanics.
 

Online alancalverd

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Yes, since a Lissajous figure is simply a plot of x = a sin ft against y = b sin (f't + p) where a and b are amplitudes, f and f' are frequencies, and p  is a phase angle, it's quite a handy tool for looking at the behavior of nonlinear systems.

There was (maybe still is) a suggestion that memristors could form useful oscillators, in which case f = f' and the Lissajous figure can show the phase relationships around the tuned circuit. No big deal: we use L f 's for synchronising all sorts of things from atomic clocks to aircraft engines. 
 

Online alancalverd

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Ah, ok...good, good!  Not a Doppler shift.  Lets examine that.  The Pound Rebka attaches a gamma ray to a speaker cone and a Doppler shift is created in the test signal of the experiment.  If we attach a laser to the speaker cone then a Doppler shift would be created in the light of the laser.  So by pointing a laser at a mirror attached to a speaker cone, and pointing a laser 'straight' at it the light shining back would be Doppler shifted...

...But the Lissajous pattern is not created like this.  Looking at the tuning fork version, the laser beam  is pointed at a mirror set up at an angle that reflects the beam to another tuning fork with a mirror on it, that then reflects the beam onto the screen.  Both tuning forks are vibrating.  Both mirrors are adding a Doppler shift to the beam of laser light.

I see your confusion. The Lissajous figure is, as I said above, just the path of a light beam, particle or whatever that moves cyclically in the x and y directions simultaneously. To get a Doppler shift you need to move the source in the z direction, i.e. along the direction of propagation of the signal, which is what P&R did. You can do this in principle with a laser and a vibrating mirror but the Doppler shift is so small that it would be lost in the normal bandwidth of the laser. The trick with P&R was to use a Mossbauer filter which has an incredibly narrow bandwidth. I'll leave the numbers up to you, as you have the Doppler equation and at least as much experience of tuning forks as I have!

True story. My first day as a physics laboratory demonstrator brought me into contact with a student  who was indeed working with (but not entirely understanding) Lissajous figures. Her experimental writeup was "different patterns were obtained by fiddling with various knobs on the oscilloscope". I gave her full marks because she had written down exactly what she did and what she saw (the essence of experimental science) whilst the other students had merely copied the method scripts that I had written and put in a few plausible numbers. A few years ago she was awarded "Physics Teacher of the Year" by the Institute of Physics.
« Last Edit: 11/10/2016 23:29:52 by alancalverd »
 

Offline Colin2B

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But I am not discussing the propagation of sound - I am discussing the distribution of resonant vibration.
Sorry, I thought for a moment you were suggesting that the standing soundwaves in the plates have discrete energies associated with their frequencies. I should have realised that you would understand  why this is meaningful for v high frequencies eg light, but not for those of sound.

The standing waves that are inherent in the plate are holding it in its non-vibrating position.  Without them the plate would not be a plate.
I assume that as above you are not talking about the standing sound waves in the plate, because those standing soundwaves which form the chladni patterns do not form and have no existence until the plate is stimulated to vibrate either by hammer, bow or frequency from a loudspeaker. The standing waves are the resonance we hear. The plate is still a plate without those standing waves, so I'm confused about which standing waves are holding the plate in its nonvibrating position.

Well actually Colin, you have explained this scenario in both ways.  You have told me that a plate has natural frequencies, and the input to the plate has no frequency, in that one blows rather than whistles across the top of a bottle to create a sound, or hits with a hammer,(bowing with a violin bow is actually much more complicated physics)....
And then you tell me, as we can quite clearly observe in the links provided, that Chladni patterns are forming on a plate and changing from one pattern to another as an input is being increased in its frequency.
That's because, as I have explained, there are a number of different ways in which the plates may be stimulated, I agree that the bow is more complicated physics, but at the level we are discussing here it is still just another way of stimulating the different modes of resonance. But the point I have been trying to make is there is no inherent wave (standing or otherwise) which combines with another input frequency to create the patterns, the patterns are only the result of the resonance, standing soundwaves and their harmonics, which superpose onto each other.
You cannot draw a parallel between the chladni patterns and the lissajou figures and argue that they are the result of 2 frequencies. What is an interesting parallel is that the more interesting lissajous figures occur when the 2 frequencies have integer relationships. This is precisely what is happening with the chladni patterns as the harmonics are all integer multiples, the difference is that whereas lissajous is made from only 2 frequencies, chladni often have at least 6 frequencies in each vibrating length, so 12 for square/rectangle.

Gravity is one of the driving forces of the compound pendulum's representation of a Lissajous figure.
Agreed, however, gravity is not an input sinewave as you stated, the gravitational force is constant.

I'm not sure whether any further input from me will help rather than hinder your quest to link schrödinger, Doppler and lissajous, because such input would be based on sound propagation rather than quantum level. For example, we could discuss how when a violin string resonates it pushes against the bow making it slip across the strings at the resonant frequency, a similar effect occurs when blowing across a tube where the vortices which initiate the resonance are forced to follow the resonant frequency, but I can't see how such facts, fascinating as they are, can help you solve your problems at a quantum level.
Best of luck in your quest.
 

Offline timey

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That is a lovely story Colin.

(Edit: Just realised that should be 'lovely story Alan' and that the next 2 paragraphs are in answer to your post that I mistakenly thought Colin had posted)

The only part of the Pound Rebka experiment that interests me is the part where they say: When the Doppler shift of the test signal was of the correct type to be 'matched and cancelled out', the gamma ray was then able to be received by the receiver.

The only part of the Lissajous patterns that interests me is the 'outward' gyrations and inertias in relation to frequency and the fact that a change in mass or a change in gravity will affect the gyration and inertia of a Lissajous pattern being created by a compound pendulum.

Let's try a different tack.  A Chladni plate pattern is caused at the points of least vibration.  Now imagine that the sand was being pushed to the points of most vibration instead.  Would this be a Lissajous pattern?

*
(Edit: and to Colin's post)

Edit: Aww, just read your latest post.  Yes blowing over a bottle top or flute is really interesting and the physics are complicated because you can change the octave of a tone by changing the shape of your mouth and the way you blow.  Much the same as the wrist and pressure action of a violin bow.  Might have to open a whole new thread for that one Colin.  I know by experience how to facilitate these effects in both flute and violin, but could learn a lot about the why of it :)...
« Last Edit: 12/10/2016 16:47:24 by timey »
 

Offline timey

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Then what is required, as I have stated in new theories, is a concept of action dilation rather than just time dilation. Then time is no longer an independent background in quantum mechanics.

Well Jeff, one is free to entertain trains of thought.  I'm not quite sure what you mean by dilated action.  The equivalence principle denotes that each reference frame must be the physical equivalent of all other reference frames.

In stating time as variable in the quantum region at the stage of the blackbody data that produced Planck's h constant in the manner that I have suggested on your new theories thread, where... as the frequency is observed to be increasing, the length of the standard second is decreased so that the Planck data curve matches the Jean Reiliegh classical curve of the ultra violet catastrophe.  Then the quantised nature of the data will become linear.

We don't need to talk in terms of quantum from that point on.

This concept does however force one into several physical cul de sacs, all of which lead to the model I describe on new theories.

But for this board:

The consequences of different reference frames running at variable rates of time gives rise to the question of simultaneity and a universally constant present. 

On the basis that it would be acceptable to discuss the concept of quantum states being multiverses comprised of copies of this universe displaying the actions of what happens on the other side of Schrödinger's partitioned box, I think I can be forgiven for discussing the possibility that variable rates of time can occur simultaneously to each other, ie: side by side, and that a logical solution to maintaining the universally constant present within this construct is that our observation of reference frames of rates of time differing from our own are time frame dependent and proportional to the difference in rate of time of observation point and rate of time at point of observation.

This concept becomes interesting when considering the Bekenstien, Hawking black hole conundrum, the second law of thermodynamics and the energy conservation law.
 

Offline jeffreyH

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If we are in reference frame A and we are observing the action of a particle in reference frame B along a line element, how is that different to observing a change in relativistic mass? It is only a change in our viewpoint of the time component that makes the difference. Isn't it time that prevents a union between QM and GR?
 

Offline timey

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Well its kind of hard for me to get into the discussion of relativistic mass without straying into New Theories territory, but if you think about the implications of matching the Planck data curve to the Jean Rayliegh classical curve by decreasing the length of a standard second as frequency is increased, then you will realise that frequencies in the lower registers may require that the standard second is increased in length.

By attributing open space with a time dilation of its own that decreases the length of a second as the gravity field decreases in strength with distance from mass...

Looking at this concept of the rate of time getting slower for 'space' in relation to clocks slowing in relative motion, you will find that the necessity for relativistic mass has been rendered redundant. The universal speed limit of the speed of light is upheld as well as the equivalence principle, in that light will always travel 299 792 458 metres per second, whatever the length of that second is, the speed you travel will always be a percentage of the reference frames speed of light as per the reference frame that is being travelled through's length of second, this percentage of the speed of light you are travelling will affect your motion related slowing of time, and that the interplay of both phenomenin in relation to each other ensures that mass never achieves the speed of light of the reference frame it is travelling through, and that it is gravity potential that is the mechanical function of frequency changes for mass travelling through space.

This then forces one to look at other means of calculating what is happening with light in the gravity field, and seek an alternate perspective concerning calculating kinetic energy with respect to relative motion related slowing of time.
« Last Edit: 12/10/2016 17:00:55 by timey »
 

Online alancalverd

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Let's sum up the Lissajous stuff. Any periodic shape can be synthesised or approximated by the addition of sine waves.   A Lissajous figure is the result of vector addition of two periodic functions at right angles. That's all there is to it.
 

Offline timey

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Great stuff Alan!

But could you please tell me what the maths of an inverse representation of a Lissajous figure are, what the maths for an inverse representation of a Chladni plate are, and if the inverse representation of each relates mathematically to the non inverse representation of the other?
 

Offline Colin2B

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Let's try a different tack.  A Chladni plate pattern is caused at the points of least vibration.  Now imagine that the sand was being pushed to the points of most vibration instead.  Would this be a Lissajous pattern?
It would still be a Chladni pattern, it wouldn't be a lissajous pattern.
Chladni patterns are just a mapping of the nodes and antinodes of a plate and it just happens that the easiest way to visualise them is by indicator gathering in the nodes, It is possible to display the antinodes pattern using a form of laser holography which shows the areas of maximum displacement as dark areas.
I can't get some maths symbols to post otherwise I would show you the maths, but lissajou patterns are the locus (movement of a point) resulting from 2 sinewaves one pushing the point in the x direction and the other in the y direction (as Alan said), whereas the wavefunction for the Chladni patterns of a rectangular plate with free edges is the product of 3 cosine functions and is stationary. Because lissajous are motions of a point and Chladni patterns are stationary it doesn't make sense to talk of one being the inverse of the other.
Interesting aside, Faraday discovered that very light material e.g. fragments of bow hair, tended to move to the antinodes drawn there by surface air currents caused by the flexing of the plate.

Edit: Aww, just read your latest post.  Yes blowing over a bottle top or flute is really interesting and the physics are complicated because you can change the octave of a tone by changing the shape of your mouth and the way you blow.  Much the same as the wrist and pressure action of a violin bow.  Might have to open a whole new thread for that one Colin.  I know by experience how to facilitate these effects in both flute and violin, but could learn a lot about the why of it :)...
Yes, overblowing has its parallel in bow technique, and it really brings out an understanding of the differences between sympathetic resonances and stimulated resonance, but as you say separate subject. Also, there is limited time and space to cover these subjects fully so it would have to be a very superficial discussion like the one we've had on Chladni where we haven't even scratched the surface.

Just as an aside before we leave this. Phonons were mentioned and these as you are aware don't play a part in chladni patterns. For others reading this, phonons propagate by longitudinal and transverse displacement of atoms in the material lattice, but because these displacements are extremely small the amplitudes of the waves are also small, in fact almost inaudible. Plate vibrations on the other hand have large displacements and can produce  very loud sounds (wouldn't have string instruments otherwise - well maybe electric bass), this is because the standing waves are caused by the plates bending and twisting rather than relying on longitudinal or transverse waves through the plate.
 

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