...These are great news. I came to nearly the same conclusion some time last year(Q=79011). I never post it, at least I am not sure about the formula (found an approximation in an cern paper about cavities if my memory is correct) and my implementation. No -3dB bandwidth needed for the calculation, its mode,volume, conductivity dependent.Based on this the Q at larger volumes is in general (mode dependent) bigger than for smaller volume. I think more energy can be stored in larger volumes.If I try to use to divide all dimensions by a factor of 10, I get a 10 times higher resonant frequency (good so far) but a Q of only 24985. Could you so kind to check this please, I can feel something may still wrong with this calculation although the number for the original dimensions fits yours very well.

The 1% error is due to this approximation:(∫ElectromagneticEnergy dV/ ∫ ElectromagneticEnergy dA) ~ ~ (ElectromagneticEnergy/ElectromagneticEnergy) (∫dV/ ∫ dA ) ~ InteriorVolume/InteriorSurfaceArea ~ π R^{2}L/(2 π R (R+L) ) ~ R/(2(1+R/L))approximating the behavior of the electromagnetic mode shape as being almost constant throughout the cavity (this approximation is pretty good for a low mode like TE012 but is expected to degrade if one considers higher modes)

Quote from: X_RaY on 01/12/2016 09:19 PM...These are great news. I came to nearly the same conclusion some time last year(Q=79011). I never post it, at least I am not sure about the formula (found an approximation in an cern paper about cavities if my memory is correct) and my implementation. No -3dB bandwidth needed for the calculation, its mode,volume, conductivity dependent.Based on this the Q at larger volumes is in general (mode dependent) bigger than for smaller volume. I think more energy can be stored in larger volumes.If I try to use to divide all dimensions by a factor of 10, I get a 10 times higher resonant frequency (good so far) but a Q of only 24985. Could you so kind to check this please, I can feel something may still wrong with this calculation although the number for the original dimensions fits yours very well.Please notice that when reducing the size to 1/10 of the original size, you calculate a Q=24,985 down from the original Q=79,011So that the scaling you calculate is:X-Ray calculated Q scaling: (79,011/24,985)/Sqrt[10] = 1.0000which goes exactly like the square root of the dimension, instead of my calculation (here: https://forum.nasaspaceflight.com/index.php?topic=39214.msg1474351#msg1474351 ) using the exact solution:Rodal calculated Q scaling: (78642.44767279371`/25104.934868706456`)/Sqrt[10] = 0.990599showing that the exact solution differs by 1% from the approximate rule of Q scaling like the square root.I justify the 1% difference between the exact solution for Q and the approximation involved in the scaling calculations for Q, as due to the approximation of the energy integral in my discussion of Q scaling:QuoteThe 1% error is due to this approximation:(∫ElectromagneticEnergy dV/ ∫ ElectromagneticEnergy dA) ~ ~ (ElectromagneticEnergy/ElectromagneticEnergy) (∫dV/ ∫ dA ) ~ InteriorVolume/InteriorSurfaceArea ~ π R^{2}L/(2 π R (R+L) ) ~ R/(2(1+R/L))approximating the behavior of the electromagnetic mode shape as being almost constant throughout the cavity (this approximation is pretty good for a low mode like TE012 but is expected to degrade if one considers higher modes)QUESTION to X-Ray; are you approximating the energy integral calculation in your Q calculation as above, and is that why your calculation results in perfect scaling of Q going like the square root of the dimension ?

Quote from: Rodal on 01/15/2016 06:55 PMQuote from: X_RaY on 01/12/2016 09:19 PM...These are great news. I came to nearly the same conclusion some time last year(Q=79011). I never post it, at least I am not sure about the formula (found an approximation in an cern paper about cavities if my memory is correct) and my implementation. No -3dB bandwidth needed for the calculation, its mode,volume, conductivity dependent.Based on this the Q at larger volumes is in general (mode dependent) bigger than for smaller volume. I think more energy can be stored in larger volumes.If I try to use to divide all dimensions by a factor of 10, I get a 10 times higher resonant frequency (good so far) but a Q of only 24985. Could you so kind to check this please, I can feel something may still wrong with this calculation although the number for the original dimensions fits yours very well.Please notice that when reducing the size to 1/10 of the original size, you calculate a Q=24,985 down from the original Q=79,011So that the scaling you calculate is:X-Ray calculated Q scaling: (79,011/24,985)/Sqrt[10] = 1.0000which goes exactly like the square root of the dimension, instead of my calculation (here: https://forum.nasaspaceflight.com/index.php?topic=39214.msg1474351#msg1474351 ) using the exact solution:Rodal calculated Q scaling: (78642.44767279371`/25104.934868706456`)/Sqrt[10] = 0.990599showing that the exact solution differs by 1% from the approximate rule of Q scaling like the square root.I justify the 1% difference between the exact solution for Q and the approximation involved in the scaling calculations for Q, as due to the approximation of the energy integral in my discussion of Q scaling:QuoteThe 1% error is due to this approximation:(∫ElectromagneticEnergy dV/ ∫ ElectromagneticEnergy dA) ~ ~ (ElectromagneticEnergy/ElectromagneticEnergy) (∫dV/ ∫ dA ) ~ InteriorVolume/InteriorSurfaceArea ~ π R^{2}L/(2 π R (R+L) ) ~ R/(2(1+R/L))approximating the behavior of the electromagnetic mode shape as being almost constant throughout the cavity (this approximation is pretty good for a low mode like TE012 but is expected to degrade if one considers higher modes)QUESTION to X-Ray; are you approximating the energy integral calculation in your Q calculation as above, and is that why your calculation results in perfect scaling of Q going like the square root of the dimension ?

.../...and using this equation, one can plot the results in completely frame-indifferent terms:deltaMass/InitialMass = function (deltaV/c , InitialVelocity/c).../...

On a similar vein to frobnicat's post above: in the EM Drive thread you noted that deceleration requires significantly different negative mass creation than acceleration from rest.But consider this: a) use your device to accelerate something to some velocity b) turn it off, so that the device now moves at constant velocity c) move your frame of reference to that inertial frame, so that the device is once again at rest d) rotate the device 180 degrees then switch it back on, accelerating in that new inertial frame (and decelerating in the original frame).It is extremely hard to see how acceleration from rest in the opposite direction can require any different consumption of anything than in the original direction.R.[modified to make more sense after posting]

Quote from: Rodal on 02/03/2016 09:49 PM.../...and using this equation, one can plot the results in completely frame-indifferent terms:deltaMass/InitialMass = function (deltaV/c , InitialVelocity/c).../...It is not clear to me in your argument why InitialVelocity/c would qualify as "frame-indifferent". Different inertial observers could agree on deltaV/c but see different values for InitialVelocity/c, and hence predict different outcome deltaMass/InitialMass. The other way around, measuring a certain deltaMass/InitialMass and a certain deltaV/c would imply one peculiar InitialVelocity/c that would hold for one privileged inertial observer and not for the others. Am I misunderstanding something ?

Quote from: RERT on 02/05/2016 08:39 AM.../...It is extremely hard to see how acceleration from rest in the opposite direction can require any different consumption of anything than in the original direction..../....../...2) Take a look at the mathematical solution. The solution function is continuous, for negative values of deltaV/c it is double-valued. The asymmetry you are addressing in your response arises because you are arbitrarily taking into account only one of the possible values for negative deltaV/c. Mathematically and for consistency you should instead take into account all possible values of a multi-valued function, when addressing symmetry of a multi-valued function.Since the solution is multi-valued I need the time to further examine it in both directions, and assess its physical significance (if any, because the notion of negative mass and energy is anything but intuitive! )..../...

.../...It is extremely hard to see how acceleration from rest in the opposite direction can require any different consumption of anything than in the original direction..../...

Quote from: Rodal on 02/05/2016 07:59 PMQuote from: RERT on 02/05/2016 08:39 AM.../...It is extremely hard to see how acceleration from rest in the opposite direction can require any different consumption of anything than in the original direction..../....../...2) Take a look at the mathematical solution. The solution function is continuous, for negative values of deltaV/c it is double-valued. The asymmetry you are addressing in your response arises because you are arbitrarily taking into account only one of the possible values for negative deltaV/c. Mathematically and for consistency you should instead take into account all possible values of a multi-valued function, when addressing symmetry of a multi-valued function.Since the solution is multi-valued I need the time to further examine it in both directions, and assess its physical significance (if any, because the notion of negative mass and energy is anything but intuitive! )..../...Following from the relativistic momentum conservation (on a single axis) I get the same expression as what you wrote for deltaMassBar as a function of deltavc and vbarc. From the initial equality (momentum conservation) to this last equality there is no need to take a square root of the equation, nor solve for solutions of second degree polynomial : so why do you say the function is double-valued, and specifically for negative deltavc ?For instance with vbarc=1/2 and deltavc=-1/4 the expression unequivocally gives deltaMassBar=sqrt(5)-1 > 0What would be an other solution ?I understand this is under construction

It is extremely hard to see how acceleration from rest in the opposite direction can require any different consumption of anything than in the original direction.

...The solution makes sense for InitialVelocity/c = 0 (γ=1) (it agrees with Bondi's momentum equation for negative mass in that case). (H. Bondi, Negative Mass in General Relativity, Rev. Mod. Phys. 29, 423;1 July 1957)...

2) Forward (Robert Forward, "Negative matter propulsion", Journal of Propulsion and Power, Vol. 6, No. 1 (1990), http://arc.aiaa.org/doi/abs/10.2514/3.23219?journalCode=jpp), and Bondi, have used similar expressions when discussing momentum conservation (https://en.wikipedia.org/wiki/Negative_mass#Runaway_motion), but they only consider the case of two bodies, one with identical absolute value of mass: one body with mass +m and another one with mass -m instead of the case being discussed here of continuous variability in mass.

It is interesting to note that, in the spaceship consisting ofpositive and negative mass elements discussed by Forward, asthe total mass of the spaceship approaches zero (M _ - M+ =~ 0)the Brownian motion of the ship due to impact of various particles will buffet it around at increasingly large velocities. Evenin a perfect vacuum, photons of cosmic background radiationwill become important if the mass is low enough. AtM _ ~ M+. the mass of the ship equals zero and any impact willapparently send it moving off at the speed or light. (ActuallyM will never precisely equal zero, as the ship will be constantlyabsorbing and emitting thermal photons.)A particle hitting a zero mass spaceship would, of course,actually hit either the positive or negative mass portion. In aship consisting of nearly equal amounts of positive andnegative mass, the center of mass can move faster than eitherof the constituent masses and will do so whenever the distancebetween the two masses changes. Unless the ship is allowed tocome apart the true motion of the ship must eventually recon·cile with the motion of the center or mass. This occurs due tothe force on the link connecting the masses. The force on thelink will cause the masses to move as described by Forward, sothat even a small initial impulse will cause very large change invelocity if the positive and negative masses are nearly equal.This fact is of use in propulsion: a very nearly zero mass spaceshipcould be propelled by a flashlight.