Quote from: StrongGR on 05/14/2015 06:00 pmI 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=830137Marco, your paper, after equation 39, has this interesting comment:Quote from: FrascaThese 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 possibleNow, 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 coneHowever (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 =<r2Definer1bar = r1 / lor2bar = r2 / loThen b = (r1bar^2 - r2bar^2)/(4 Log[r2bar/r1bar])Define:r1bar = r2bar/c (where c >= 1 since r1bar =< r2bar )bb = b / (r2bar^2)thenbb = (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] = - 0.342542Limit[bb, c -> 2] = - 0.270505Limit[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 ~ r2And 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 stepNow, since the maximum (absolute magnitude) value of b is b = (- 1/2) (r2bar^2) and we know that r2bar = r2 / lothen the maximum value of b isb = (- 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 < lothen it necessarily follows that the maximum possible value of b is much, much smaller than 1b << 1we 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.

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

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.

...Dear Jose,Thanks a lot for spending some time on my calculations. You tried with b but did you check for a? Also l0 contains the density of energy of the electromagnetic field (the constant U0 squared). In any case, I am aware, as my colleagues, that general relativity could not be enough. My aim is to explore this approach to evaluate the magnitude of the effects in play. If such an effect would be confirmed by independent measurements at other NASA labs, as stated in the news, it would mean that we have to cope with something real new. It would be a great result. Otherwise, Baez and others will have reasons to laugh.

Quote from: deuteragenie on 05/16/2015 09:57 pmQuote from: Rodal on 05/16/2015 08:22 pmQuote from: deuteragenie on 05/16/2015 08:01 pmQuote from: Rodal on 05/16/2015 05:20 pm4) Also this link may be helpful, giving a table of the Xmn and X'mn to 15 digits accuracyhttp://wwwal.kuicr.kyoto-u.ac.jp/www/accelerator/a4/besselroot.htmlxTry: Wolfram alpha Thanks. That link ( https://www.wolframalpha.com/input/?i=besseljzero[1%2C1 ) only gives X_{mn} which are only useful for TM modesIs 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 ln67OK, 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

Quote from: Rodal on 05/16/2015 08:22 pmQuote from: deuteragenie on 05/16/2015 08:01 pmQuote from: Rodal on 05/16/2015 05:20 pm4) Also this link may be helpful, giving a table of the Xmn and X'mn to 15 digits accuracyhttp://wwwal.kuicr.kyoto-u.ac.jp/www/accelerator/a4/besselroot.htmlxTry: Wolfram alpha Thanks. That link ( https://www.wolframalpha.com/input/?i=besseljzero[1%2C1 ) only gives X_{mn} which are only useful for TM modesIs 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

Quote from: deuteragenie on 05/16/2015 08:01 pmQuote from: Rodal on 05/16/2015 05:20 pm4) Also this link may be helpful, giving a table of the Xmn and X'mn to 15 digits accuracyhttp://wwwal.kuicr.kyoto-u.ac.jp/www/accelerator/a4/besselroot.htmlxTry: Wolfram alpha Thanks. That link ( https://www.wolframalpha.com/input/?i=besseljzero[1%2C1 ) only gives X_{mn} which are only useful for TM modesIs there a way that Wolfram Alpha can give X'_{mn} which are needed for TE modes ?

Quote from: Rodal on 05/16/2015 05:20 pm4) Also this link may be helpful, giving a table of the Xmn and X'mn to 15 digits accuracyhttp://wwwal.kuicr.kyoto-u.ac.jp/www/accelerator/a4/besselroot.htmlxTry: Wolfram alpha

4) Also this link may be helpful, giving a table of the Xmn and X'mn to 15 digits accuracyhttp://wwwal.kuicr.kyoto-u.ac.jp/www/accelerator/a4/besselroot.htmlx

http://www.ibtimes.co.uk/emdrive-warp-drive-are-two-different-things-nasas-still-working-emdrive-1501268Interesting comments from Shawyer are attached:

Roger Shawyer, the British scientist who invented the highly controversial electromagnetic space propulsion technology called EmDrive, has said Nasa's work is encouraging but still far behind many private firms working on it already."Obviously I'm very happy for Nasa, they're having great fun, but it's not really real science," Shawyer told IBTimes UK in an exclusive interview."Obviously I'm very happy for Nasa, they're having great fun, but it's not really real science," Shawyer told IBTimes UK in an exclusive interview.

I agree, this statement:Quote from: ibtimes ukhas said Nasa's work is encouraging but still far behind many private firms working on it already."Obviously I'm very happy for Nasa, they're having great fun, but it's not really real science," Shawyer told IBTimes UK in an exclusive interview.is a very, very strange, not objective comment (for Shawyer to be quoted in http://www.ibtimes.co.uk/nasa-validates-emdrive-roger-shawyer-says-aerospace-industry-needs-watch-out-1499141 as saying that "NASA is just having great fun, but the experiments at NASA are not really science" in Shawyer's opinion)

has said Nasa's work is encouraging but still far behind many private firms working on it already."Obviously I'm very happy for Nasa, they're having great fun, but it's not really real science," Shawyer told IBTimes UK in an exclusive interview.

....I have plotted with Maple the function bb(c)=(1-c^2)/(4*c^2*ln(c)) and I have found the figure I enclose. There is no maximum whatsoever. This curve has an asymptote at increasing c and another for c going to zero. This is what I meant saying that geometry comes to rescue. I am currently working out all this with Mathematica and I will update my draft as soon as possible with realistic values. Of course, my take remains the same as yours, if the effect exists Einstein theory could not be enough.

Quote from: TheTraveller on 05/17/2015 12:03 pmhttp://www.ibtimes.co.uk/emdrive-warp-drive-are-two-different-things-nasas-still-working-emdrive-1501268Interesting comments from Shawyer are attached:Aside from such declarations, what I regret for is a dismissal a priori of an object really easy to test, with just few bucks.

At the time of cold fusion scientific community was much more open minded. Important labs tried to repeat the experiment with the help of Fleishmann himself.

Now, there are nor secrets neither patents at stake. It seems like we learnt to behave like Bellarmino at Galileo time where it was far too easy to look at the sky with the scope, but hard to admit to have been wrong and dismiss a lot of acquired "knowledge".

We know from direct measurements that EM fields behave in a specific manner, down to parts per trillion. Momentum which is carried by an electromagnetic field can be measured directly as the field strength, with far greater precision. This knowledge puts a very low upper bound on the net momentum that EM fields can acquire and exchange with the cavity.

Quote from: StrongGR on 05/17/2015 01:19 pm....I have plotted with Maple the function bb(c)=(1-c^2)/(4*c^2*ln(c)) and I have found the figure I enclose. There is no maximum whatsoever. This curve has an asymptote at increasing c and another for c going to zero. This is what I meant saying that geometry comes to rescue. I am currently working out all this with Mathematica and I will update my draft as soon as possible with realistic values. Of course, my take remains the same as yours, if the effect exists Einstein theory could not be enough.Take a gander at my proof http://forum.nasaspaceflight.com/index.php?topic=36313.msg1375279#msg1375279. I had explicitly stated:r1bar = r2bar/c (where c >= 1 since r1bar =< r2bar )One has to assume that either r1 is smaller than r2 or that r2 is smaller than r1.In your paper ( http://forum.nasaspaceflight.com/index.php?action=dlattach;topic=36313.0;attach=830137 ) you assumed that r1 is smaller than r2, both in the integral and also in the expression showing the term (r2-r1)/h, in your equations 34 and 38 of your paper.in order for r1 to be equal or less than r2, c must be equal to or greater than 1.Instead you have plotted c for values smaller than 1.Your plot shows values for c in the wrong range. c is defined to have values from 1 to Infinity, instead of from 0 to 1 as you plotted.The values of c smaller than 1 you plotted are in conflict with the assumption that r1 is smaller than r2 (which was explicitly defined). To be consistent with the assumption that r1 is smaller than r2, you should plot values of c equal to or greater than 1.

Quote from: StrongGR on 05/17/2015 01:04 pmQuote from: TheTraveller on 05/17/2015 12:03 pmhttp://www.ibtimes.co.uk/emdrive-warp-drive-are-two-different-things-nasas-still-working-emdrive-1501268Interesting comments from Shawyer are attached:Aside from such declarations, what I regret for is a dismissal a priori of an object really easy to test, with just few bucks. It's not a-priori. It's a-posteriori. We didn't know a-priori that it does not work. We know from direct measurements that EM fields behave in a specific manner, down to parts per trillion. Momentum which is carried by an electromagnetic field can be measured directly as the field strength, with far greater precision. This knowledge puts a very low upper bound on the net momentum that EM fields can acquire and exchange with the cavity. Hence the knowledge that his drive does not work is a-posteriori knowledge. If his force was in low nano or pico newtons range, it would have been an a-priori assumption.QuoteAt the time of cold fusion scientific community was much more open minded. Important labs tried to repeat the experiment with the help of Fleishmann himself. Scientific community's open mindedness towards a claim depends mainly to the claim itself.QuoteNow, there are nor secrets neither patents at stake. It seems like we learnt to behave like Bellarmino at Galileo time where it was far too easy to look at the sky with the scope, but hard to admit to have been wrong and dismiss a lot of acquired "knowledge".How can you compare a person who received a grant from the British government to Galileo?

....Your assumption is not needed as this function retains his sign independently on your choice. In any case, there are not extrema and this function runs to zero on the right and runs to minus infinity on the left. You can verify yourself that the function is symmetric under the exchange r1<->r2.

Does the Flight Thruster have a slightly concave top and convex bottom? Would appear so from the gaps.Enhanced the photo as much as I can for those wishing to try to extract dimensions as this photo is better that the original as it has no distortion.If we can find the dimension<M

Quote from: StrongGR on 05/17/2015 01:53 pm....Your assumption is not needed as this function retains his sign independently on your choice. In any case, there are not extrema and this function runs to zero on the right and runs to minus infinity on the left. You can verify yourself that the function is symmetric under the exchange r1<->r2.Sorry, but you cannot consistently first write equations where r1 is assumed to be less than r2 (as per your paper) and now proceed to show a graph in conflict with YOUR assumption that r1 was smaller than r2. That is inconsistent. Incorrect results follow from inconsistencies in formulation.I suggest that you:1) make a plot of the geometry of the truncated cone you have in mind: clearly showing what you define to be r1, r2, h, and the z and r coordinates2) explicitly show what is the optimal geometry of the truncated cone, as per your paper.Further discussion without you clearly defining your geometrical variables is moot and can lead nowhere, as your paper assumes r1 <r2 and now you show plots for r1 > r2, in strong conflict with your assumed geometry and equations