 ### Author Topic: EM Drive Developments - related to space flight applications - Thread 2  (Read 2597008 times)

#### Notsosureofit

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##### Re: EM Drive Developments - related to space flight applications - Thread 2
« Reply #3500 on: 05/16/2015 04:19 pm »
Anyone seen this little tidbit?
http://phys.org/news/2015-05-newton-law-broken.html
Non-equiibrium thermodynamics

I have to find my old copy of the book by S. R. De Groot, P. Mazur and other such books I have in boxes

and see whether we can derive a force for the EM Drive based on Onsager's relations Still looking for my copy as well !  (old cavity work was on transport properties)

The system in that paper is not isolated from the environment. For EM Drive this is not true. You have to cope with a closed system and to understand why third law appears to be violated, if the effect is confirmed

Yup !

Do you remember if DeGroot mentioned photons ?

That book is rather old. I just gave a cursory look and found nothing about photons. You should rather consider the case for electromagnetic radiation. What could make the thing resemble that of a non-isolated system is leakage of radiation out of the cavity. I have to guess that one has built it reducing such losses. The idea to look at gravitational effects, started by Minotti's paper, is because in this case there is not a boundary due to the cavity and system appears to be open (space-time is everywhere). So, the third law would be saved by the expulsion of gravitational momentum. The reason why people in the community of scientists did not consider the case is the smallness of the gravity with respect to all other effects and so, one should not expect it to account for such a measurement.

As you can see from my crude dispersion argument, the g value (of the photons) is much larger than one might naively expect.

"In the accelerated frame of reference w/ the acceleration, g = (c^2/(2*L*f^2))*(c/(2*Pi))^2*X^2*((1/Rs^2)-(1/Rb^2)) [for this waveguide-like approximation]."

I missed something. Did you write a paper about? Please, let me know as I have not followed this discussion from the start.

FYI

Cleanup and detypo of the take on applying the Equivalence Principle.

The proposition that dispersion caused by an accelerating frame of reference implied an accelerating frame of reference caused by a dispersive cavity resonator. (to 1st order using massless, perfectly conducting cavity)

Starting with the expressions for the frequency of a cylindrical RF cavity:

f = (c/(2*Pi))*((X/R)^2+((p*Pi)/L)^2)^.5

For TM modes, X = X[sub m,n] = the n-th zero of the m-th Bessel function.
[1,1]=3.83, [0,1]=2.40, [0,2]=5.52 [1,2]=7.02, [2,1]=5.14, [2,2]=8.42, [1,3]=10.17, etc.

and for TE modes, X = X'[subm,n] = the n-th zero of the derivative of the m-th Bessel function.
[0,1]=3.83, [1,1]=1.84, [2,1]=3.05, [0,2]=7.02, [1,2]=5.33, [1,3]=8.54, [0,3]=10.17, [2,2]=6.71, etc.

Rotate the dispersion relation of the cavity into doppler frame to get the Doppler shifts, that is to say, look at the dispersion curve intersections of constant wave number instead of constant frequency.

delta(f) = (1/(2*f))*(c/(2*Pi))^2*X^2*((1/Rs^2)-(1/Rb^2))

and from there the expression for the acceleration g from:

g = (c^2/L)*(delta(f)/f) such that:

g = (c^2/(2*L*f^2))*(c/(2*Pi))^2*X^2*((1/Rs^2)-(1/Rb^2))

Using the "weight" of the photon in the accelerated frame from:

"W" = (h*f/c^2)*g =>  "W" = T = (h/L)*delta(f)

gives thrust per photon:

T = (h/(2*L*f))*(c/(2*pi))^2*X^2*((1/Rs^2)-(1/Rb^2))

If the number of photons is (P/hf)*(Q/2*pi*f) then:

NT = P*Q*(1/(4*pi*L*f^3))*(c/(2*pi))^2*X^2*((1/Rs^2)-(1/Rb^2))

This does fit (as far as I've gotten) the concept of a self-accelerating Dirac wavepacket (which does conserve momentum).
---------------------------------------------------------------------------------------------------------------------------
Q is the effective Q due to phase shift etc.
« Last Edit: 05/20/2015 05:05 pm by Notsosureofit »

#### StrongGR

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##### Re: EM Drive Developments - related to space flight applications - Thread 2
« Reply #3501 on: 05/16/2015 04:22 pm »
...Gads I love the internet and wished it was there when I was going to school. I'm not ready to throw the slab of idea and thought meat into the shark pool yet but thanks for the links!!! Kudos!
Not sure about that...wonder how much individual thought process, creativity and imagination is overwhelmed by the huge amount of information available from the Internet nowadays.  Many scientists think that it may have the opposite effect on individual intelligence and thought process as the mind needs as much or more exercise (to produce connections of neurons, etc.) as human muscles.  With so many "answers" freely available at a click of one's fingers, there is much less exercise of one's brain Hopefully not a future of people thinking that Physics and Engineering is just a question of getting answers from a huge Internet cookbook of recipes and from black-box computer software instead of using mathematics and experiments to find out how Nature operates...

As an active researcher, internet appears essential for my work. Journals accept submissions online and can be read online. Arxiv is an internet site without which I cannot work at all. Extensive bibliographic researches can be easily accomplished and I need them to write down a paper. Besides, there are blogs, emails and forums like this. All together this made my life as a researcher really easy. Of course, it is often really difficult to tell what is good from what is bad but this is all what happens in everyday life and does not seem so difficult for well trained people.

#### StrongGR

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##### Re: EM Drive Developments - related to space flight applications - Thread 2
« Reply #3502 on: 05/16/2015 04:45 pm »
Anyone seen this little tidbit?
http://phys.org/news/2015-05-newton-law-broken.html
Non-equiibrium thermodynamics

I have to find my old copy of the book by S. R. De Groot, P. Mazur and other such books I have in boxes

and see whether we can derive a force for the EM Drive based on Onsager's relations Still looking for my copy as well !  (old cavity work was on transport properties)

The system in that paper is not isolated from the environment. For EM Drive this is not true. You have to cope with a closed system and to understand why third law appears to be violated, if the effect is confirmed

Yup !

Do you remember if DeGroot mentioned photons ?

That book is rather old. I just gave a cursory look and found nothing about photons. You should rather consider the case for electromagnetic radiation. What could make the thing resemble that of a non-isolated system is leakage of radiation out of the cavity. I have to guess that one has built it reducing such losses. The idea to look at gravitational effects, started by Minotti's paper, is because in this case there is not a boundary due to the cavity and system appears to be open (space-time is everywhere). So, the third law would be saved by the expulsion of gravitational momentum. The reason why people in the community of scientists did not consider the case is the smallness of the gravity with respect to all other effects and so, one should not expect it to account for such a measurement.

As you can see from my crude dispersion argument, the g value (of the photons) is much larger than one might naively expect.

"In the accelerated frame of reference w/ the acceleration, g = (c^2/(2*L*f^2))*(c/(2*Pi))^2*X^2*((1/Rs^2)-(1/Rb^2)) [for this waveguide-like approximation]."

I missed something. Did you write a paper about? Please, let me know as I have not followed this discussion from the start.

FYI

Cleanup and detypo of the take on applying the Equivalence Principle.

The proposition that dispersion caused by an accelerating frame of reference implied an accelerating frame of reference caused by a dispersive cavity resonator. (to 1st order using massless, perfectly conducting cavity)

Starting with the expressions for the frequency of an RF cavity:

f = (c/(2*Pi))*((X/R)^2+((p*Pi)/L)^2)^.5

For TM modes, X = X[sub m,n] = the n-th zero of the m-th Bessel function.
[1,1]=3.83, [0,1]=2.40, [0,2]=5.52 [1,2]=7.02, [2,1]=5.14, [2,2]=8.42, [1,3]=10.17, etc.

and for TE modes, X = X'[subm,n] = the n-th zero of the derivative of the m-th Bessel function.
[0,1]=3.83, [1,1]=1.84, [2,1]=3.05, [0,2]=7.02, [1,2]=5.33, [1,3]=8.54, [0,3]=10.17, [2,2]=6.71, etc.

Rotate the dispersion relation of the cavity into doppler frame to get the Doppler shifts, that is to say, look at the dispersion curve intersections of constant wave number instead of constant frequency.

df = (1/(2*f))*(c/(2*Pi))^2*X^2*((1/Rs^2)-(1/Rb^2))

and from there the expression for the acceleration g from:

g = (c^2/L)*(df/f) such that:

g = (c^2/(2*L*f^2))*(c/(2*Pi))^2*X^2*((1/Rs^2)-(1/Rb^2))

Using the "weight" of the photon in the accelerated frame from:

"W" = (h*f/c^2)*g =>  "W" = T = (h/L)*df

gives thrust per photon:

T = (h/(2*L*f))*(c/(2*pi))^2*X^2*((1/Rs^2)-(1/Rb^2))

If the number of photons is (P/hf)*(Q/2*pi*f) then:

NT = P*Q*(1/(4*pi*L*f^3))*(c/(2*pi))^2*X^2*((1/Rs^2)-(1/Rb^2))

This does fit (as far as I've gotten) the concept of a self-accelerating Dirac wavepacket (which does conserve momentum).
---------------------------------------------------------------------------------------------------------------------------
Yes the search function is terrible

df is delta f not the differential,  Q is the effective Q due to phase shift etc.

I have put your text in a LaTeX format and enclose it here. But the question is: Where does your acceleration come from?

#### Notsosureofit

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##### Re: EM Drive Developments - related to space flight applications - Thread 2
« Reply #3503 on: 05/16/2015 04:57 pm »
Anyone seen this little tidbit?
http://phys.org/news/2015-05-newton-law-broken.html
Non-equiibrium thermodynamics

I have to find my old copy of the book by S. R. De Groot, P. Mazur and other such books I have in boxes

and see whether we can derive a force for the EM Drive based on Onsager's relations Still looking for my copy as well !  (old cavity work was on transport properties)

The system in that paper is not isolated from the environment. For EM Drive this is not true. You have to cope with a closed system and to understand why third law appears to be violated, if the effect is confirmed

Yup !

Do you remember if DeGroot mentioned photons ?

That book is rather old. I just gave a cursory look and found nothing about photons. You should rather consider the case for electromagnetic radiation. What could make the thing resemble that of a non-isolated system is leakage of radiation out of the cavity. I have to guess that one has built it reducing such losses. The idea to look at gravitational effects, started by Minotti's paper, is because in this case there is not a boundary due to the cavity and system appears to be open (space-time is everywhere). So, the third law would be saved by the expulsion of gravitational momentum. The reason why people in the community of scientists did not consider the case is the smallness of the gravity with respect to all other effects and so, one should not expect it to account for such a measurement.

As you can see from my crude dispersion argument, the g value (of the photons) is much larger than one might naively expect.

"In the accelerated frame of reference w/ the acceleration, g = (c^2/(2*L*f^2))*(c/(2*Pi))^2*X^2*((1/Rs^2)-(1/Rb^2)) [for this waveguide-like approximation]."

I missed something. Did you write a paper about? Please, let me know as I have not followed this discussion from the start.

FYI

Cleanup and detypo of the take on applying the Equivalence Principle.

The proposition that dispersion caused by an accelerating frame of reference implied an accelerating frame of reference caused by a dispersive cavity resonator. (to 1st order using massless, perfectly conducting cavity)

Starting with the expressions for the frequency of an RF cavity:

f = (c/(2*Pi))*((X/R)^2+((p*Pi)/L)^2)^.5

For TM modes, X = X[sub m,n] = the n-th zero of the m-th Bessel function.
[1,1]=3.83, [0,1]=2.40, [0,2]=5.52 [1,2]=7.02, [2,1]=5.14, [2,2]=8.42, [1,3]=10.17, etc.

and for TE modes, X = X'[subm,n] = the n-th zero of the derivative of the m-th Bessel function.
[0,1]=3.83, [1,1]=1.84, [2,1]=3.05, [0,2]=7.02, [1,2]=5.33, [1,3]=8.54, [0,3]=10.17, [2,2]=6.71, etc.

Rotate the dispersion relation of the cavity into doppler frame to get the Doppler shifts, that is to say, look at the dispersion curve intersections of constant wave number instead of constant frequency.

df = (1/(2*f))*(c/(2*Pi))^2*X^2*((1/Rs^2)-(1/Rb^2))

and from there the expression for the acceleration g from:

g = (c^2/L)*(df/f) such that:

g = (c^2/(2*L*f^2))*(c/(2*Pi))^2*X^2*((1/Rs^2)-(1/Rb^2))

Using the "weight" of the photon in the accelerated frame from:

"W" = (h*f/c^2)*g =>  "W" = T = (h/L)*df

gives thrust per photon:

T = (h/(2*L*f))*(c/(2*pi))^2*X^2*((1/Rs^2)-(1/Rb^2))

If the number of photons is (P/hf)*(Q/2*pi*f) then:

NT = P*Q*(1/(4*pi*L*f^3))*(c/(2*pi))^2*X^2*((1/Rs^2)-(1/Rb^2))

This does fit (as far as I've gotten) the concept of a self-accelerating Dirac wavepacket (which does conserve momentum).
---------------------------------------------------------------------------------------------------------------------------
Yes the search function is terrible

df is delta f not the differential,  Q is the effective Q due to phase shift etc.

I have put your text in a LaTeX format and enclose it here. But the question is: Where does your acceleration come from?

The accelerated frame or reference in which the asymmetry of the cavity is balanced out.  That is as though the cavity was cylindrical in this case.  Notice you would probably need the coordinate changes to do this as a proper derivation.  As we said, that might have to constitute a (mathematical) proof.

#### StrongGR

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##### Re: EM Drive Developments - related to space flight applications - Thread 2
« Reply #3504 on: 05/16/2015 05:06 pm »
Anyone seen this little tidbit?
http://phys.org/news/2015-05-newton-law-broken.html
Non-equiibrium thermodynamics

I have to find my old copy of the book by S. R. De Groot, P. Mazur and other such books I have in boxes

and see whether we can derive a force for the EM Drive based on Onsager's relations Still looking for my copy as well !  (old cavity work was on transport properties)

The system in that paper is not isolated from the environment. For EM Drive this is not true. You have to cope with a closed system and to understand why third law appears to be violated, if the effect is confirmed

Yup !

Do you remember if DeGroot mentioned photons ?

That book is rather old. I just gave a cursory look and found nothing about photons. You should rather consider the case for electromagnetic radiation. What could make the thing resemble that of a non-isolated system is leakage of radiation out of the cavity. I have to guess that one has built it reducing such losses. The idea to look at gravitational effects, started by Minotti's paper, is because in this case there is not a boundary due to the cavity and system appears to be open (space-time is everywhere). So, the third law would be saved by the expulsion of gravitational momentum. The reason why people in the community of scientists did not consider the case is the smallness of the gravity with respect to all other effects and so, one should not expect it to account for such a measurement.

As you can see from my crude dispersion argument, the g value (of the photons) is much larger than one might naively expect.

"In the accelerated frame of reference w/ the acceleration, g = (c^2/(2*L*f^2))*(c/(2*Pi))^2*X^2*((1/Rs^2)-(1/Rb^2)) [for this waveguide-like approximation]."

I missed something. Did you write a paper about? Please, let me know as I have not followed this discussion from the start.

FYI

Cleanup and detypo of the take on applying the Equivalence Principle.

The proposition that dispersion caused by an accelerating frame of reference implied an accelerating frame of reference caused by a dispersive cavity resonator. (to 1st order using massless, perfectly conducting cavity)

Starting with the expressions for the frequency of an RF cavity:

f = (c/(2*Pi))*((X/R)^2+((p*Pi)/L)^2)^.5

For TM modes, X = X[sub m,n] = the n-th zero of the m-th Bessel function.
[1,1]=3.83, [0,1]=2.40, [0,2]=5.52 [1,2]=7.02, [2,1]=5.14, [2,2]=8.42, [1,3]=10.17, etc.

and for TE modes, X = X'[subm,n] = the n-th zero of the derivative of the m-th Bessel function.
[0,1]=3.83, [1,1]=1.84, [2,1]=3.05, [0,2]=7.02, [1,2]=5.33, [1,3]=8.54, [0,3]=10.17, [2,2]=6.71, etc.

Rotate the dispersion relation of the cavity into doppler frame to get the Doppler shifts, that is to say, look at the dispersion curve intersections of constant wave number instead of constant frequency.

df = (1/(2*f))*(c/(2*Pi))^2*X^2*((1/Rs^2)-(1/Rb^2))

and from there the expression for the acceleration g from:

g = (c^2/L)*(df/f) such that:

g = (c^2/(2*L*f^2))*(c/(2*Pi))^2*X^2*((1/Rs^2)-(1/Rb^2))

Using the "weight" of the photon in the accelerated frame from:

"W" = (h*f/c^2)*g =>  "W" = T = (h/L)*df

gives thrust per photon:

T = (h/(2*L*f))*(c/(2*pi))^2*X^2*((1/Rs^2)-(1/Rb^2))

If the number of photons is (P/hf)*(Q/2*pi*f) then:

NT = P*Q*(1/(4*pi*L*f^3))*(c/(2*pi))^2*X^2*((1/Rs^2)-(1/Rb^2))

This does fit (as far as I've gotten) the concept of a self-accelerating Dirac wavepacket (which does conserve momentum).
---------------------------------------------------------------------------------------------------------------------------
Yes the search function is terrible

df is delta f not the differential,  Q is the effective Q due to phase shift etc.

I have put your text in a LaTeX format and enclose it here. But the question is: Where does your acceleration come from?

The accelerated frame or reference in which the asymmetry of the cavity is balanced out.  That is as though the cavity was cylindrical in this case.  Notice you would probably need the coordinate changes to do this as a proper derivation.  As we said, that might have to constitute a (mathematical) proof.

You have a cavity standing still. You claim there is a Doppler shift in the modes. What is the origin of this Doppler shift? I cannot see any way to get "accelerated" photons here as you claim they are.

#### Notsosureofit

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##### Re: EM Drive Developments - related to space flight applications - Thread 2
« Reply #3505 on: 05/16/2015 05:12 pm »
Yes the search function is terrible

df is delta f not the differential,  Q is the effective Q due to phase shift etc.

I have put your text in a LaTeX format and enclose it here. But the question is: Where does your acceleration come from?
[/quote]

Thank you !  I'll look up LaTex (new to me)   How does one edit the pdf ?  ( df should be delta f, for example)

Its the (taking the extremes of the) equivalent wavelength change along the cavity axis of the resonant frequency in the rest frame.  It should really be the result of an integral over that length to get a more general case ( included dielectrics for example )

#### StrongGR

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##### Re: EM Drive Developments - related to space flight applications - Thread 2
« Reply #3506 on: 05/16/2015 05:18 pm »
Yes the search function is terrible

df is delta f not the differential,  Q is the effective Q due to phase shift etc.

I have put your text in a LaTeX format and enclose it here. But the question is: Where does your acceleration come from?

Thank you !  I'll look up LaTex (new to me)   How does one edit the pdf ?  ( df should be delta f, for example)

Its the (taking the extremes of the) equivalent wavelength change along the cavity axis of the resonant frequency in the rest frame.  It should really be the result of an integral over that length to get a more general case ( included dielectrics for example )
[/quote]

You are welcome. But what is the moving frame? I see a cavity that is still. No way to get a frequency changed (gravity is an exception).

#### Rodal

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##### Re: EM Drive Developments - related to space flight applications - Thread 2
« Reply #3507 on: 05/16/2015 05:20 pm »
...
I have put your text in a LaTeX format and enclose it here. But the question is: Where does your acceleration come from?
Watch out for errors on the transcription to LaTeX pdf http://forum.nasaspaceflight.com/index.php?action=dlattach;topic=36313.0;attach=830511

1) equation (1) reads

2c/Pi

c/(2*Pi)

2) Also there is an error in Eq. (5)

T=(h/c)*delta f

T=(h/L)*delta f

3) "c" in Notsosureofit's post stands for

speed-of-light-in-medium = c/Sqrt[μr εr ]

where c is the speed of light in vacuum

4) Also this link may be helpful, giving a table of the Xmn and  X'mn to 15 digits accuracy

http://wwwal.kuicr.kyoto-u.ac.jp/www/accelerator/a4/besselroot.htmlx

« Last Edit: 05/16/2015 06:08 pm by Rodal »

#### StrongGR

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##### Re: EM Drive Developments - related to space flight applications - Thread 2
« Reply #3508 on: 05/16/2015 05:22 pm »
Quote
Thank you !  I'll look up LaTex (new to me)   How does one edit the pdf ?  ( df should be delta f, for example)

Its the (taking the extremes of the) equivalent wavelength change along the cavity axis of the resonant frequency in the rest frame.  It should really be the result of an integral over that length to get a more general case ( included dielectrics for example )

I fixed that delta f.
« Last Edit: 05/16/2015 05:24 pm by StrongGR »

#### StrongGR

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##### Re: EM Drive Developments - related to space flight applications - Thread 2
« Reply #3509 on: 05/16/2015 05:24 pm »
...
I have put your text in a LaTeX format and enclose it here. But the question is: Where does your acceleration come from?
Watch out for errors on the transcription to LaTeX pdf.

For example, first equation (1) reads 2c/Pi instead of c/(2*Pi)

http://forum.nasaspaceflight.com/index.php?action=dlattach;topic=36313.0;attach=830511

Fixed. Thanks.

#### Acryte

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##### Re: EM Drive Developments - related to space flight applications - Thread 2
« Reply #3510 on: 05/16/2015 05:28 pm »
Today i did the first test with the Emdrive (microwave oven magnetron and cooper frustum)
The setup (magnetron and frusum) was suspended in a pendulum.
I applied power for 40 Seconds with no visible thrust. Tomorrow will will try again with the magnetron on the small side. You have any suggestion for what should be the distance from the small side?
After this i will adjust the power to the filament of magnetron and the frequency.
To fine adjust the frequency i thought i can put 2 coils over the magnetron magnets to modify the magnetic field.
My website;
http://www.masinaelectrica.com/emdrive-independent-test/

Don't burn your house down. Your setup looks inadequately safe.

#### aero

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##### Re: EM Drive Developments - related to space flight applications - Thread 2
« Reply #3511 on: 05/16/2015 05:36 pm »
...
I have put your text in a LaTeX format and enclose it here. But the question is: Where does your acceleration come from?
Watch out for errors on the transcription to LaTeX pdf.

For example, first equation (1) reads 2c/Pi instead of c/(2*Pi)

http://forum.nasaspaceflight.com/index.php?action=dlattach;topic=36313.0;attach=830511

Fixed. Thanks.

It would be helpful to mark revision number both inside the file and in the title. Maybe with a date inside.

And thanks, the LaTeX pdf is so much easier to read even if LaTeX can't be inserted directly into the text of this thread. ( I suppose it could be inserted or attached as an image? But that's not necessary.)
« Last Edit: 05/16/2015 05:41 pm by aero »
Retired, working interesting problems

#### StrongGR

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##### Re: EM Drive Developments - related to space flight applications - Thread 2
« Reply #3512 on: 05/16/2015 05:42 pm »
...
I have put your text in a LaTeX format and enclose it here. But the question is: Where does your acceleration come from?
Watch out for errors on the transcription to LaTeX pdf.

For example, first equation (1) reads 2c/Pi instead of c/(2*Pi)

http://forum.nasaspaceflight.com/index.php?action=dlattach;topic=36313.0;attach=830511

Fixed. Thanks.

It would be helpful to mark revision number both inside the file and in the title. Maybe with a date inside.

Yes, knowing LaTeX, all free from internet (editor included), it is possible to accomplish all the tasks you are asking for. I did this just because it is really difficult to read ascii formulas and I would like to understand the argument by Notsureofit. These forums do not foresee to use LaTeX and this can make it overall complicated.

#### WarpTech

##### Re: EM Drive Developments - related to space flight applications - Thread 2
« Reply #3513 on: 05/16/2015 06:29 pm »
Anyone seen this little tidbit?
http://phys.org/news/2015-05-newton-law-broken.html
Non-equiibrium thermodynamics

I have to find my old copy of the book by S. R. De Groot, P. Mazur and other such books I have in boxes

and see whether we can derive a force for the EM Drive based on Onsager's relations Still looking for my copy as well !  (old cavity work was on transport properties)

The system in that paper is not isolated from the environment. For EM Drive this is not true. You have to cope with a closed system and to understand why third law appears to be violated, if the effect is confirmed

Yup !

Do you remember if DeGroot mentioned photons ?

That book is rather old. I just gave a cursory look and found nothing about photons. You should rather consider the case for electromagnetic radiation. What could make the thing resemble that of a non-isolated system is leakage of radiation out of the cavity. I have to guess that one has built it reducing such losses. The idea to look at gravitational effects, started by Minotti's paper, is because in this case there is not a boundary due to the cavity and system appears to be open (space-time is everywhere). So, the third law would be saved by the expulsion of gravitational momentum. The reason why people in the community of scientists did not consider the case is the smallness of the gravity with respect to all other effects and so, one should not expect it to account for such a measurement.

As you can see from my crude dispersion argument, the g value (of the photons) is much larger than one might naively expect.

"In the accelerated frame of reference w/ the acceleration, g = (c^2/(2*L*f^2))*(c/(2*Pi))^2*X^2*((1/Rs^2)-(1/Rb^2)) [for this waveguide-like approximation]."

I missed something. Did you write a paper about? Please, let me know as I have not followed this discussion from the start.

FYI

Cleanup and detypo of the take on applying the Equivalence Principle.

The proposition that dispersion caused by an accelerating frame of reference implied an accelerating frame of reference caused by a dispersive cavity resonator. (to 1st order using massless, perfectly conducting cavity)

Starting with the expressions for the frequency of an RF cavity:

f = (c/(2*Pi))*((X/R)^2+((p*Pi)/L)^2)^.5

For TM modes, X = X[sub m,n] = the n-th zero of the m-th Bessel function.
[1,1]=3.83, [0,1]=2.40, [0,2]=5.52 [1,2]=7.02, [2,1]=5.14, [2,2]=8.42, [1,3]=10.17, etc.

and for TE modes, X = X'[subm,n] = the n-th zero of the derivative of the m-th Bessel function.
[0,1]=3.83, [1,1]=1.84, [2,1]=3.05, [0,2]=7.02, [1,2]=5.33, [1,3]=8.54, [0,3]=10.17, [2,2]=6.71, etc.

Rotate the dispersion relation of the cavity into doppler frame to get the Doppler shifts, that is to say, look at the dispersion curve intersections of constant wave number instead of constant frequency.

df = (1/(2*f))*(c/(2*Pi))^2*X^2*((1/Rs^2)-(1/Rb^2))

and from there the expression for the acceleration g from:

g = (c^2/L)*(df/f) such that:

g = (c^2/(2*L*f^2))*(c/(2*Pi))^2*X^2*((1/Rs^2)-(1/Rb^2))

Using the "weight" of the photon in the accelerated frame from:

"W" = (h*f/c^2)*g =>  "W" = T = (h/L)*df

gives thrust per photon:

T = (h/(2*L*f))*(c/(2*pi))^2*X^2*((1/Rs^2)-(1/Rb^2))

If the number of photons is (P/hf)*(Q/2*pi*f) then:

NT = P*Q*(1/(4*pi*L*f^3))*(c/(2*pi))^2*X^2*((1/Rs^2)-(1/Rb^2))

This does fit (as far as I've gotten) the concept of a self-accelerating Dirac wavepacket (which does conserve momentum).
---------------------------------------------------------------------------------------------------------------------------
Yes the search function is terrible

df is delta f not the differential,  Q is the effective Q due to phase shift etc.

I have put your text in a LaTeX format and enclose it here. But the question is: Where does your acceleration come from?

It comes from "work" done by the waves on the copper, i.e. EM induction, which slows down the wave and steals it's momentum. It mimics gravity in this regard, because photons lose energy going "up".

I'm still working on a slightly different version of this. My time has been severely limited this past week however.

Todd D.

#### deltaMass

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##### Re: EM Drive Developments - related to space flight applications - Thread 2
« Reply #3514 on: 05/16/2015 07:41 pm »
Perhaps MathJax would enhance this forum? StackExchange uses it to great advantage.

#### deuteragenie

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##### Re: EM Drive Developments - related to space flight applications - Thread 2
« Reply #3515 on: 05/16/2015 08:01 pm »
4) Also this link may be helpful, giving a table of the Xmn and  X'mn to 15 digits accuracy
http://wwwal.kuicr.kyoto-u.ac.jp/www/accelerator/a4/besselroot.htmlx

Try:

Wolfram alpha
« Last Edit: 05/16/2015 08:03 pm by deuteragenie »

#### deuteragenie

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##### Re: EM Drive Developments - related to space flight applications - Thread 2
« Reply #3516 on: 05/16/2015 09:57 pm »
4) Also this link may be helpful, giving a table of the Xmn and  X'mn to 15 digits accuracy
http://wwwal.kuicr.kyoto-u.ac.jp/www/accelerator/a4/besselroot.htmlx

Try:

Wolfram alpha
Thanks.  That link ( https://www.wolframalpha.com/input/?i=besseljzero[1%2C1  ) only gives Xmn which are only useful for TM modes

Is there a way that Wolfram Alpha can give  X'mn which are needed for TE modes ?

I don't think so. But I believe you have Mathemagica:
http://www.me.rochester.edu/courses/ME201/webexamp/derbesszer.pdf - see ln67

#### Rodal

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##### Re: EM Drive Developments - related to space flight applications - Thread 2
« Reply #3517 on: 05/16/2015 10:33 pm »
4) Also this link may be helpful, giving a table of the Xmn and  X'mn to 15 digits accuracy
http://wwwal.kuicr.kyoto-u.ac.jp/www/accelerator/a4/besselroot.htmlx

Try:

Wolfram alpha
Thanks.  That link ( https://www.wolframalpha.com/input/?i=besseljzero[1%2C1  ) only gives Xmn which are only useful for TM modes

Is there a way that Wolfram Alpha can give  X'mn which are needed for TE modes ?

I don't think so. But I believe you have Mathemagica:
http://www.me.rochester.edu/courses/ME201/webexamp/derbesszer.pdf - see ln67
OK, yes, that's helpful to the people that have Mathematica.  They can program that function ( notice that X'mn is not a buil-in function in Mathematica yet).  To those that don't have Mathematica, that's the great usefulness of this link http://wwwal.kuicr.kyoto-u.ac.jp/www/accelerator/a4/besselroot.htmlx: it gives X'mn to 15 digits accuracy and it is accessible to everyone.

The Kyoto (Japan) University link only goes to m=10, n=5 though, so if anyone knows of a link going to a higher quantum number (n > 5) , please post it, as it may be helpful to people just using Excel.  (In case anyone is interested in investigating super-high mode shapes for n > 5 « Last Edit: 05/16/2015 10:49 pm by Rodal »

#### Rodal

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##### Re: EM Drive Developments - related to space flight applications - Thread 2
« Reply #3518 on: 05/17/2015 12:12 am »
I am back with an updated draft after some terrible news around about NASA dismissing these researches. They should not as, otherwise, it could happen as with Galilei having his detractors even not trying to look in the telescope, just dismissing on faith.

I have analysed the case of the frustum and the results appear to be striking. One must admit that geometry comes to rescue not just general relativity. For this particular geometry the cavity can be made susceptible to gravitational effects if your choice of the two radii of the cavity is smart enough. This is something to be confirmed yet, just my theoretical result, but shocking anyway.

As usual, any comment is very welcome.

http://forum.nasaspaceflight.com/index.php?action=dlattach;topic=36313.0;attach=830137

Marco, your paper, after equation 39, has this interesting comment:

Quote from: Frasca
These equations appear rather interesting as, by a proper choice of parameters, one can make a gravitational effect more or less relevant in the physics of the problem. It is the case to say that geometry comes to rescue.

Just what this "proper choice of parameters" is, it is not spelled out in your paper, so I will follow my interpretation of your equation.

It appears (see proof below) that in order to maximize the constants "a" and "b" in Equation 39, we want to have:

1) r1 as close as possible to r2
2) r2 as large as possible

Now, if this is correct, this is a rather peculiar, surprising geometry: it says that the axial length of the truncated cone should be close to zero while the radius should be as large as possible, in other words, the geometry should be as close to an almost perfect cylinder (small cone angle) with very short axial length, and with flat faces. In other words, this optimized geometry, according to Eq. 39 in your paper seems to be much closer to Cannae's device.
And it is actually not far from the geometry presently used by Dr. White. It is certainly not the geometry of a cone However  (see below), we also conclude that the maximum possible values of the constants b (and also a) are extremely small and hence they are negligible unless the EM Drive happens to be near a magnestar.

The "lo" constant is a really huge number (unless, as you state the EM Drive happens to be located next to the field of a magnetar, which is certainly not the case, I might add, since the closest magnetar SGR 1806-20 is located about 50,000 light-years away from Earth ).

PROOF:

Taking (from Eq. 39 in your paper)

b = (r1^2 - r2^2)/(4 (lo^2) Log[r2/r1])

Assuming r1 =<r2

Define

r1bar = r1 / lo
r2bar = r2 / lo

Then

b = (r1bar^2 - r2bar^2)/(4 Log[r2bar/r1bar])

Define:

r1bar = r2bar/c  (where c >= 1 since r1bar =< r2bar )

bb = b / (r2bar^2)

then

bb = (1 - c^2)/(4 (c^2) Log[c])

Limit[bb, c -> 1] = -  0.5  (this corresponds to the maximum possible value of r1, r1 ~ r2 )

Limit[bb, c -> 1.5] = -  0.342542

Limit[bb, c -> 2] = - 0.270505

Limit[bb, c -> Infinity] = 0  (this corresponds to the minimum possible value of r1, r1 -> 0 )

Then it follows that the maximum absolute value of bb  (bb= - 0.5) occurs at c = 1 , at r1 ~ r2
And to maximize b we must maximize r2bar, and therefore maximize r2, since b =  bb (r2bar^2)= (- 1/2) (r2bar^2)

The proof for "a' is similar but it involves an extra step

Now, since the maximum (absolute magnitude) value of b is b = (- 1/2) (r2bar^2) and we know that
r2bar = r2 / lo
then the maximum value of b is

b = (- 1/2) ((r2 / lo)^2)

but since we know that lo is a huge number (unless the EM Drive is next to a magnestar) and feasible values of r2 are such that

r2 < lo

then it necessarily follows that the maximum possible value of b is much, much smaller than 1

b  << 1

we conclude that the maximum possible values of b (and also a) are extremely small and hence they are negligible unless the EM Drive happens to be near a magnetar http://en.wikipedia.org/wiki/Magnetar. « Last Edit: 05/17/2015 07:37 am by Rodal »

#### WarpTech

##### Re: EM Drive Developments - related to space flight applications - Thread 2
« Reply #3519 on: 05/17/2015 01:28 am »
...
FYI

Cleanup and detypo of the take on applying the Equivalence Principle.

The proposition that dispersion caused by an accelerating frame of reference implied an accelerating frame of reference caused by a dispersive cavity resonator. (to 1st order using massless, perfectly conducting cavity)

Starting with the expressions for the frequency of an RF cavity:

f = (c/(2*Pi))*((X/R)^2+((p*Pi)/L)^2)^.5

....

Rotate the dispersion relation of the cavity into doppler frame to get the Doppler shifts, that is to say, look at the dispersion curve intersections of constant wave number instead of constant frequency.

df = (1/(2*f))*(c/(2*Pi))^2*X^2*((1/Rs^2)-(1/Rb^2))

...

...

It comes from "work" done by the waves on the copper, i.e. EM induction, which slows down the wave and steals it's momentum. It mimics gravity in this regard, because photons lose energy going "up".

I'm still working on a slightly different version of this. My time has been severely limited this past week however.

Todd D.

@Notsosureofit, I just realized why I was perplexed by df, though I understood what you meant I could not derive it. Now I think I see what you have is;

df = (f(Rs)^2 - f(Rb)^2/2f

What I would do is take the derivative wrt L, where R is a fn of L.

df = - (1/f) * (c/2pi)^2 * ((X^2 / R^3) * dR + ((p*pi)^2 / L^3) * dL)

What is going on here is something akin to the Relativistic energy equation:

p^2 = E^2 - m^2   (for c = 1)

The dispersion relation correlates to;

1/lambda_g^2 = 1/lambda_0^2 - 1/lambda_c^2

Where _g is for guide, _0 is for free space and _c is for cut-off wavelengths.

Taking the derivative, lambda_0 is a constant and that term goes to zero. It is then easy to see that the relationship between the guide wavelength and the cut-off are equal and opposite, and how it accelerates.

Standing waves have E and B out of phase 90 deg. These fields can do no work. They are there to store and accumulate energy, like an amplifier.

Where work is being done is when E < m and p is imaginary. That imaginary component adds a 90 degree phase shift, that aligns E & B so that work can be done on the frustum. For those Electrical Engineers out there, it adjusts the Power Factor from 0 to 1.

This occurs each time waves travel back toward the small end. Their momentum is absorbed. The force will be the rate, dp/dt.

p = sqrt(E^2 - m^2), which will be imaginary. We want to maximize this asymmetrically.

This is why my gut instinct tells me that trying to optimize Q and resonance in the same device required to generate attenuation is counter productive. The amplifier and the attenuator should be separate systems.

Todd D.
« Last Edit: 05/17/2015 01:36 am by WarpTech »

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