Author Topic: EM Drive Developments - related to space flight applications - Thread 3  (Read 1876306 times)

Offline Rodal

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....Incorrect if you are disputing the units of the Poynting vector as I described them. Are you?
For non-elementary problems in Physics  the fundamental quantity to derive forces is the stress.   The equations of equilibrium for non-uniform stress are formulated in terms of stress and not forces.
I don't care. I described the units of the Poynting vector and you replied with "incorrect". You either need to say why I'm incorrect or retract your statement.

I replied to the "Correct" affirmation to WWS statement that <<  Just because you see a net, time-averaged Poynting vector doesn't mean you can conclude anything about any unbalanced forces.  >> in

...Maybe I misunderstand you, but I'm going to keep saying it for good measure:  The Poynting vector, S, is NOT a force.  Just because you see a net, time-averaged Poynting vector doesn't mean you can conclude anything about any unbalanced forces.  An constant unbalanced force  could only be generated if you saw that Poynting vector continuously and endlessly growing. 

This is the link to my previous comment on this issue.   
Correct, as a cursory glance at the units of the Poynting vector will show. It has units of power per unit area, whereby the area refers to the plane orthogonal to the direction of the power flux

which is an over-statement, as the time-averaged (over an integer number of periods) of the Poynting vector divided by the speed of light in free space is the radiation pressure exerted by an electromagnetic wave on the surface of a target, if the wave is completely absorbed by the target .

Stress = <Poynting vector averaged over integer number of periods> / c

An increasing with time Poynting vector is not needed in that case, all that is needed is for the average over integer number of periods to be greater than zero, in order to have radiation pressure when the wave is absorbed by the target.

This formula has been verified experimentally multiple times since 1900 when the wave is absorbed by the target.

Equilibirum equations for non-uniform states of stress (which is the case in the EM Drive) are formulated in terms of stresses and not in terms of forces.  Forces result from surface integration of stress tensor components.
« Last Edit: 07/05/2015 05:32 PM by Rodal »

Offline wallofwolfstreet

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....Incorrect if you are disputing the units of the Poynting vector as I described them. Are you?
For non-elementary problems in Physics  the fundamental quantity to derive forces is the stress.   The equations of equilibrium for non-uniform stress are formulated in terms of stress and not forces.
I don't care. I described the units of the Poynting vector and you replied with "incorrect". You either need to say why I'm incorrect or retract your statement.

The statement <<  Just because you see a net, time-averaged Poynting vector doesn't mean you can conclude anything about any unbalanced forces.  >> in

...Maybe I misunderstand you, but I'm going to keep saying it for good measure:  The Poynting vector, S, is NOT a force.  Just because you see a net, time-averaged Poynting vector doesn't mean you can conclude anything about any unbalanced forces.  An constant unbalanced force  could only be generated if you saw that Poynting vector continuously and endlessly growing. 

This is the link to my previous comment on this issue.   
Correct, as a cursory glance at the units of the Poynting vector will show. It has units of power per unit area, whereby the area refers to the plane orthogonal to the direction of the power flux

is an over-statement, as the time-averaged (over an integer number of periods) of the Poynting vector divided by the speed of light in free space is the radiation pressure exerted by an electromagnetic wave on the surface of a target, if the wave is completely absorbed by the target .

Stress = <Poynting vector averaged over integer number of periods> / c 

this formula has been verified experimentally multiple times since 1900 when the wave is absorbed by the target

Completely disagree.  My statement << Just because you see a net, time-averaged Poynting vector doesn't mean you can conclude anything about any unbalanced forces. >> is exactly correct. 

In specific instances, can a poynting vector be used in an equation to derive force?  Yes.  As you yourself say, its's elementary.  The photon rocket and solar sail are both instances where the poynting vector clearly has a relation with force.

In general however, you can never conclude force from a Poynting vector alone. Look at the attached figure.  There is a "time-averaged (over an integer number of periods)" net Poynting vector.  NO force.  Put a piece of metal sheet between the battery and the resistor.  Is there radiation pressure acting there?  No, obviously there isn't.  Poynting vectors and radiation pressure don't have a 1-1 correspondence.

That figure alone should be enough to convince you that in general, the Poynting vector can not be reduced to a force.  Clearly there is some difference between a solar sail being "pushed" by the poynting flux from the sun, and the lack of force on a metal sheet caught in the poynting flux of a simple DC circuit.   

Offline Rodal

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...

Completely disagree.  My statement << Just because you see a net, time-averaged Poynting vector doesn't mean you can conclude anything about any unbalanced forces. >> is exactly correct. 

In specific instances, can a poynting vector be used in an equation to derive force?  Yes.  As you yourself say, its's elementary.  The photon rocket and solar sail are both instances where the poynting vector clearly has a relation with force.

In general however, you can never conclude force from a Poynting vector alone. Look at the attached figure.  There is a "time-averaged (over an integer number of periods)" net Poynting vector.  NO force.  Put a piece of metal sheet between the battery and the resistor.  Is there radiation pressure acting there?  No, obviously there isn't.  Poynting vectors and radiation pressure don't have a 1-1 correspondence.

That figure alone should be enough to convince you that in general, the Poynting vector can not be reduced to a force.  Clearly there is some difference between a solar sail being "pushed" by the poynting flux from the sun, and the lack of force on a metal sheet caught in the poynting flux of a simple DC circuit.   

Yes, we disagree.  I would not presume, as you are doing, what other analysts can or cannot conclude from their knowledge of a subject. 

___
PS: this is a state of non-uniform stress, one of the first steps in analyzing it is to deal with stresses and not forces.  Poynting vectors are fluxes: power over surface area, therefore the discussion should be about stresses and not forces.
« Last Edit: 07/05/2015 05:43 PM by Rodal »

Offline wallofwolfstreet

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...

Completely disagree.  My statement << Just because you see a net, time-averaged Poynting vector doesn't mean you can conclude anything about any unbalanced forces. >> is exactly correct. 

In specific instances, can a poynting vector be used in an equation to derive force?  Yes.  As you yourself say, its's elementary.  The photon rocket and solar sail are both instances where the poynting vector clearly has a relation with force.

In general however, you can never conclude force from a Poynting vector alone. Look at the attached figure.  There is a "time-averaged (over an integer number of periods)" net Poynting vector.  NO force.  Put a piece of metal sheet between the battery and the resistor.  Is there radiation pressure acting there?  No, obviously there isn't.  Poynting vectors and radiation pressure don't have a 1-1 correspondence.

That figure alone should be enough to convince you that in general, the Poynting vector can not be reduced to a force.  Clearly there is some difference between a solar sail being "pushed" by the poynting flux from the sun, and the lack of force on a metal sheet caught in the poynting flux of a simple DC circuit.   

Yes, we disagree.  I would not presume, as you are doing, what other analysts can or cannot conclude from their knowledge of a subject. 

___
PS: this is a state of non-uniform stress, one of the first steps in analyzing it is to deal with stresses and not forces.  Poynting vectors are fluxes: power over surface area, therefore the discussion should be about stresses and not forces.

What is it that I am "presuming"?

Offline Rodal

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...What is it that I am "presuming"?

Quote from: wallofwolfstreet
Just because you see a net, time-averaged Poynting vector doesn't mean you can conclude anything about any unbalanced forces.

If that applies to what you can conclude at the present time, please don't presume that it applies to what others may conclude (based on their knowledge), or even to what you may conclude in the future.
« Last Edit: 07/05/2015 05:52 PM by Rodal »

Offline wallofwolfstreet

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...What is it that I am "presuming"?

Quote from: wallofwolfstreet
Just because you see a net, time-averaged Poynting vector doesn't mean you can conclude anything about any unbalanced forces.

If that applies to what you can conclude, please don't presume that it applies to what others may conclude (based on their knowledge).

I don't understand.  Are you saying that there is a force on a piece of material placed between the battery and resistor in a simple DC circuit?

If you answer no to that question, than we must be in agreement.  There is a Poynting vector, there isn't a force.  QED, my statement << Just because you see a net, time-averaged Poynting vector doesn't mean you can conclude anything about any unbalanced forces. >> is correct. 

If you answerd yes to the question, well I guess you know something I don't.     

Offline mwvp

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This guys think another way

http://arxiv.org/abs/0708.3519

:)


"Photons inside a waveguide as massive particles"

What a great paper! Several days ago I started to attempt to answer Rodal's question, 'If it's ratcheting, what is it ratcheting against?'. I concluded what the authors of the paper concluded (bottom).

Yes indeed, the standing wave composed of forward/reverse EM waves at C are stationary in the vacuum. The frustrum, with its currents, is the rotor of a motor, and ratchets against the stationary vacuum EM modes- the stator.

Good to hear Traveller confirm the ratchet/negative inertial resistance priniciple of operation per Shawyer. Dirty little secret he was obfuscating!

Physicists can make things so complicated and obfuscated with axions and QM and saying "evanescent" instead of capacitive/inductive reactance, and what-not.

I think, "Linear Sagnac Ratchet" would be a fitting description of the EM Drive.

pg. 11

"...the vacuum medium as a whole is always resting with respect to all inertial observers. In
other words, the relative velocity between the vacuum medium and an arbitrary inertial
observer cannot be measured (i.e., it is an unobservable quantity), such that one can think it
always vanishes. On the other hand, consider that the velocity of light in vacuum is
invariant with respect to all inertial observers...
...
the velocity of light in vacuum ( c = 1) and the velocity of the vacuum medium
( u = 0 ) are only two genuine velocities in our universe, they are invariant constants for all
inertial frames of reference; all other velocities are the apparent (or average) velocities of
massless fields moving in a zigzag manner.
...
a massive particle corresponds to massless fields
captured in a bag in the form of standing waves, where the bag consists of the vacuum
medium which is unobservable, lossless, and resting with respect to any observer. For an
observer moving with a uniform velocity ( vg ) relative to the standing waves inside the bag,
the bag formed by the vacuum medium keeps resting and appears as a resting waveguide
extending infinitely along the direction of motion, and the massive particle corresponds to
the massless fields moving in a zigzag manner along the waveguide, with the group velocity
of vg . The rest mass of the particle depends on the bag’s sizes and topological structure,
and may also depend on the properties of the vacuum medium inside the bag, as well as the
properties of the massless fields captured in the bag."


BTW,

Are the Meep simulations and Poynting vectors being recorder during the period when the frustrum is fluxing-up, or after several microseconds when an equilibrium between power-input vs. dissipation has been reached?
« Last Edit: 07/05/2015 06:33 PM by mwvp »

Offline Rodal

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...What is it that I am "presuming"?

Quote from: wallofwolfstreet
Just because you see a net, time-averaged Poynting vector doesn't mean you can conclude anything about any unbalanced forces.

If that applies to what you can conclude, please don't presume that it applies to what others may conclude (based on their knowledge).

I don't understand.  Are you saying that there is a force on a piece of material placed between the battery and resistor in a simple DC circuit?

If you answer no to that question, than we must be in agreement.  There is a Poynting vector, there isn't a force.  QED, my statement << Just because you see a net, time-averaged Poynting vector doesn't mean you can conclude anything about any unbalanced forces. >> is correct. 

If you answerd yes to the question, well I guess you know something I don't.     
We are discussing and analyzing the EM Drive in this thread.   I have not "thrown my hat into the ring" (to borrow your phrase) with regards to whether it is an artifact or not, and if not why it works.  My discussion above concerns overstatements.  Your initial post on the Poynting vector  was helpful to people in another thread that may have read a formal conclusion to a proof, when instead we are still working the problem and we have made important advances in understanding the problem (namely disproving Greg Egan's contention that there is a zero time averaged Poynting vector in the EM Drive over an integer number of periods). 

However interpreting your statement as saying that nobody can conclude anything about balance of forces, given a Poynting vector field is a bridge too far, an overstatement over which (if that is what you intended) we disagree.

Concerning the EM Drive, I look forward to the upcoming presentation of the EM Drive experiments by Tajmar, Fiedler et.al, at The Technische Universität Dresden, which, from what I gather have been conducted under vacuum and show very low force/InputPower readings but still exceeding (by less than a few dozen or so times) the force/InputPower of a perfectly collimated photon rocket.

It will be interesting to see whether the experiments at The Technische Universität Dresden can be interpreted as a scientific nullification of the claims made by Yang  and Shawyer, since Yang and Shawyer claim over 1,000 to 10,000 times greater force/InputPower than what Tajmar measured, but still a validation of something interesting going on with the EM Drive that exceeds the output of a photon rocket.

If so, this will be now two prestigious institutions (NASA and The Technische Universität Dresden, as Tajmar works also in contracts with Airbus Defence and Space) that will have tested in vacuum and obtained results (which although lower by a factor or 1,000 to 10,000 times the huge claims of Shawyer and Yang who only tested in ambient pressure and hence may be the result of thermal convection and other gas effects) points towards something interesting happening in the EM Drive.
« Last Edit: 07/05/2015 06:46 PM by Rodal »

Online kdhilliard

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Following discussions with Roger Shawyer, I now understand why using a scale to measure EMDrive Force generation is a waste of time.

Hey Traveller,

When I asked about this back in May, you replied that Shawyer has "placed them directly on scales, hung them from springs above scales, used balance beams with scales, plus he used a rotary air bearing system to show true acceleration."  Are you now saying that Shawyer does not believe that his scales based experiments returned useful results, or are you saying that they were more complicated then they appeared, and were not simply a shielded drive sitting on a scale?

~Kirk

Offline A_M_Swallow

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Following discussions with Roger Shawyer, I now understand why using a scale to measure EMDrive Force generation is a waste of time.

1) Basically when you sit the EMDrive on your scale based measurement system, it will be non moving.

2) Once you fill the frustum with microwaves it will be in IDLE mode waiting for some external Force to move/vibrate it slightly big end to small end.

3) Once that happens the EMDrive enters MOTOR mode and starts to build up an external Force. That Force will push against the measurement system and achieve a slight amount of room to slightly accelerate.

4) However soon after what ever compressive movement that was there is now gone, the EMDrive stops moving and flips back into IDLE mode and shutting off Force generation.

{snip}

Please do a control test in which the EMDrive is given the external force whilst switched off. Friction should stop the apparatus after a few seconds. If the EMDrive generated force is less than the air resistance and bearing fiction it will take longer to stop. Hopefully the generated force is greater than the friction so the apparatus keeps spinning.

Offline ludkokanta

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If you setup a sealed system to give an impulse that would provoke particles to move what would seem faster then the speed of light in classical mechanics, consequential reaction would be smaller then the impulse and a part of the impulse would "externalize".

This is just my guess.

What happens on the level of quantum physics will probably be the last thing we find out on the topic. If we knew quantum physics this well then we wouldn't even need an experiment. However, empirical is still the primary method and everything more specific is also way more abstract.

My working prototype uses a different approach, has better power to weight ratio and will probably be the one to withstand the test of time.


Offline Rodal

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...Are the Meep simulations and Poynting vectors being recorder during the period when the frustrum is fluxing-up, or after several microseconds when an equilibrium between power-input vs. dissipation has been reached?
The RF feed is constantly ON during EM Drive experiments.  As Dr. Notsosureofit stated, an equlilibrium may not be reached, at least as a rigorous definition of equilibirum is concerned.  Even for rather simple coupled systems of nonlinear differential equations it is difficult to define a "steady state" in a rigorous sense.  This issue needs further exploration as the Meep analysis already shows a nonlinear response with time, at least so far in the ~325 times steps that the FD solution has been marched forward.
« Last Edit: 07/05/2015 07:01 PM by Rodal »

Offline wallofwolfstreet

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....

I am aware we are discussing the emdrive, how could I not be?  Not sure what that statement was supposed to convey. 

This isn't a complicated issue.  Yes or no, is there a force on a piece of material placed between the battery and resistor of a DC circuit?   

There is an even easier way to get to the heart of this issue.  Let's say I have a Poynting vector field, S(r,t) over all of R3, generated by some "device".  I take the integral of this vector field over all of R3 and over some time interval (t1,t2), and discover it does not vanish over any interval of time, i.e. there is a net Poynting vector.

For any of the time intervals in question, is there a force on the device?  Not just a net force, but any forces/stresses at all? 

If the answer is "impossible to tell with the information given", then my statement is not incorrect or even an "overstatement" as you say.  You see a Poynting vector, but you can't conclude anything about the distribution of forces.  This is what I said, and you seem to think it is wrong.     

Offline ThinkerX

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Doctor McCulloch made a comment on the MEEP results:

Quote
I applaud the NSF people for the work: it's interesting (though they don't give the magnitudes in their plots) but I've been revising Poynting vectors & I don't believe they are going to move the cavity: the consequences still have to obey conservation of momentum as usual (so nothing to do with MiHsC).

In general, I'm wary of the tendency in physics to use complex computer models (eg: thermal explanation of Pioneer, dark matter) to explain things, it is so seductive but the details are hidden and it can give the right answer for the wrong reasons. How many free parameters are in the model? Have they tested it on cavity data?

Offline cej

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Again...!

http://www.nature.com/ncomms/2014/140306/ncomms4300/full/ncomms4300.html
"Momentum and spin represent fundamental dynamic properties of quantum particles and fields. In particular, propagating optical waves (photons) carry momentum and longitudinal spin determined by the wave vector and circular polarization, respectively. Here we show that exactly the opposite can be the case for evanescent optical waves. A single evanescent wave possesses a spin component, which is independent of the polarization and is orthogonal to the wave vector. Furthermore, such a wave carries a momentum component, which is determined by the circular polarization and is also orthogonal to the wave vector. We show that these extraordinary properties reveal a fundamental Belinfante’s spin momentum, known in field theory and unobservable in propagating fields. We demonstrate that the transverse momentum and spin push and twist a probe Mie particle in an evanescent field. This allows the observation of ‘impossible’ properties of light and of a fundamental field-theory quantity, which was previously considered as ‘virtual’."

What is the relationship between the angular momentum (SAM and/or OAM) and the equation for the energy of a photon, E = hf? Does this equation assume zero (or net zero) angular momentum via the correspondence principle?

Is that abstract essentially saying that the observed linear momentum of a photon (in an evanescent optical wave) is a function of its spin instead of (or in addition to) its wavelength?

Offline dustinthewind

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DERIVATION OF THRUST FROM A TAPERED WAVEGUIDE PHOTON ROCKET

I updated the equations to make the derivation clearer.  In these equations, z is the axis of the cone, theta is the half-angle, E is energy, M and m are mass terms, and the rest is just wave vectors in a waveguide.

You can see that if the cone were not tapered there would be no dependence on z, the last term in the force equation would not exist. If it were not starting in a waveguide, the phase velocity would be c. Then we have a "flashlight" photon rocket. This is a different animal. This Force only applies when the big end is OPEN! Closed, all bets are off, but this explains where the tremendous thrust to power ratios are coming from.

FYI: This is the foundation of the paper I'm writing. Now you have the "tech" right in front of you while I try to put this in "writing". Have at it! Can't wait to see what sort of designs you come up with. 8)
Todd

Would you say then that the light being reflected down the tapered cavity is trying to set up a standing wave but the magnetic modes keep accelerating downward as they expand?  In doing so they are continuously pushed out of phase with the currents in the tapered cavities side walls?  These out of phase currents in the side walls then interact with the light destructively attenuating further reflections and instead converting it into a force and a little heat? 

On the other hand I have a tendency to think of light as the sum of the local magnetic field propagating at the speed of light.  What I mean by this is if you take the Biot-Savart law (magnetic field) for a single charge (1/r^2 behavior) and manipulate it into having a velocity and of the speed of light then integrating it from infinity to "r" you get the electric field of light (1/r behavior).  I'll attache what I have for this (see page 121 to 123).  In other words, a form of dynamic magnetic fields in motion from other currents in the cavity that will propagate to infinity are interacting with other local cavity currents currents in a non-linear way. 

The 1/r^3 dependence comes after integrating the Biot-Savart law in a loop which corresponds to a dipole shift in charge due to time space distortion by relativity (see page 13 - 19). 

Would I be wrong in assuming what appears to be propulsion is just the interaction of the light with the currents in the cavity? 
« Last Edit: 07/05/2015 08:15 PM by dustinthewind »

Offline ThinkerX

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Quote
If you setup a sealed system to give an impulse that would provoke particles to move what would seem faster then the speed of light in classical mechanics, consequential reaction would be smaller then the impulse and a part of the impulse would "externalize".

This is just my guess.

What happens on the level of quantum physics will probably be the last thing we find out on the topic. If we knew quantum physics this well then we wouldn't even need an experiment. However, empirical is still the primary method and everything more specific is also way more abstract.

My working prototype uses a different approach, has better power to weight ratio and will probably be the one to withstand the test of time.

Would you care to add your results to the wiki?  (link on page 1, post 1 of this thread)

Offline ludkokanta

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My apparatus is different and I am not disclosing until IP is protected and published. I can say for sure that I could get 3 grams from a 100 gram apparatus. Now I get 2 from 500 grams with common materials.

Offline Rodal

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Doctor McCulloch made a comment on the MEEP results:

Quote
I applaud the NSF people for the work: it's interesting (though they don't give the magnitudes in their plots) but I've been revising Poynting vectors & I don't believe they are going to move the cavity: the consequences still have to obey conservation of momentum as usual (so nothing to do with MiHsC).

In general, I'm wary of the tendency in physics to use complex computer models (eg: thermal explanation of Pioneer, dark matter) to explain things, it is so seductive but the details are hidden and it can give the right answer for the wrong reasons. How many free parameters are in the model? Have they tested it on cavity data?
The comment about free parameters was previously made by him also in reference to "analysis with a large number of Finite Elements" by JPL of the Pioneer anomaly or something to that effect.  I'm sorry to say this but this is either a poorly phrased statement or a misunderstanding of FD and FE methods, as of course the number of finite elements or finite difference grid points are not free-parameters.

The repeated statements by a number of people in this regard reveals to me a lack of understanding of numerical methods in Physics.  John Von Neumann, arguably the best mathematical physicist of the 20th century, was far away from such an attitude, on the contrary, Von Neumann pioneered the full use of these methods (although Von Neumann was much better at pure math than people criticizing the use of these methods).  Ditto with Courant, Lax and many others.  I also never met this attitude against numerical methods towards understanding problems at MIT or other institutions when discussing difficult problems.  For all my theses I did simultaneously experiments, theoretical work and numerical modeling.  Certainly the use of numerical methods was paramount for the Atomic Bomb Project, for NASA going to the Moon, and it is of paramount importance for any serious problem. 

It almost feels like some people confuse numerical methods with Computer Aided Design, with black boxes where people just get an output given an input.  It is well known that one of the advantages of Meep (that's the use of it at MIT) is that it is an open code: you can write your own constitutive equations.   

So the argument that "Meep" cannot do this or that is a canard.  It would be like somebody saying "you cannot do this or that with Mathematica".  How can somebody say that about an open code?

Even when I was using ADINA (Finite Element code) for cutting edge projects for the Department of Defense as far back as 1981, I (and other team members) were writing our own constitutive equation subroutines (and certainly post-processing) instead of relying on ADINA's built-in models.   Ditto for ABAQUS.  Ditto for PETROS (Finite Difference Analysis for Predicting Large Elastic-Visco-Plastic Transient Deformations of Variable Thickness Soft-Bonded Thin Shells to explosive loading) .

Now I have seen people saying that "I have Mathematica running and ready" but no evidence that they use Mathematica for anything else than as a powerful calculator, or that they run other's people's programs (instead of writing their own programs).  To state that one cannot do this or that with Meep or Mathematica would be like saying you cannot do this or that with Fortran or C.

I am not surprised that people may not be that knowledgeable about numerical methods.  What I am surprised is about the sermonizing about what others may or may not be able to do.
« Last Edit: 07/06/2015 12:32 AM by Rodal »

Offline Ricvil

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http://forum.nasaspaceflight.com/index.php?action=dlattach;topic=37642.0;attach=1040027


DERIVATION OF THRUST FROM A TAPERED WAVEGUIDE PHOTON ROCKET

I updated the equations to make the derivation clearer.  In these equations, z is the axis of the cone, theta is the half-angle, E is energy, M and m are mass terms, and the rest is just wave vectors in a waveguide.

You can see that if the cone were not tapered there would be no dependence on z, the last term in the force equation would not exist. If it were not starting in a waveguide, the phase velocity would be c. Then we have a "flashlight" photon rocket. This is a different animal. This Force only applies when the big end is OPEN! Closed, all bets are off, but this explains where the tremendous thrust to power ratios are coming from.

FYI: This is the foundation of the paper I'm writing. Now you have the "tech" right in front of you while I try to put this in "writing". Have at it! Can't wait to see what sort of designs you come up with. 8)
Todd


Where the first equation comes from, and what is "Xmn" ???

It comes from the circular resonator equation. Where, Xmn  denotes the n-th zero of the m-th cylindrical Bessel function, used for TM modes. For TE modes, X'mn  denotes the n-th  zero of the derivative of the m-th cylindrical Bessel function. Rather than a longitudinal p-mode, I've shown it as kz as a traveling wave.
Todd

Your first equation ( the definition of the wave vector K), is wrong.
It establishes a erroneous z dependence on K.
The K will depend at most on total geometry( including the dimensions of cavity), medium constitutive relations (mu and epslon), and boundary conditions, which defines de modes inside the sctruture, and thus the modes cut off frequencys.
« Last Edit: 07/05/2015 09:04 PM by Ricvil »

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