Wanted to examine the inventor's theory of operation. We spent a lot of time discussing Shawyer's theory, but Fetta didn't get much discussion. Probably because his paper is paywalled. Also @Wembley, thanks for the patents. There's plenty of free info there.

Break:

Neat I can hit zero every half wavelength with any frequency like this. (top)

Or if I mirror the diagonal of a cylinder, creating a cone I get pretty close sometimes after adjusted down 1 degree. (bottom)

http://goo.gl/jF8ZJB (shortened link to ebay)

Found these puppies. Emailed the seller to see what the dimensions are and to see if they are the manufacturer and whether they do custom sizes. Might be cheaper and less headache than buying a copper sheet.

Something very nice about the exact solution I obtained for the frequencies and mode shapes of a truncated cone, or for a cone, is that it implicitly, automatically, incorporates the condition of cutoff frequencies. The exact solution for a cylinder in Wikipedia (

http://en.wikipedia.org/wiki/Microwave_cavity#Cylindrical_cavity) does not explicitly refer to the need for the condition of cutoff frequencies, and therefore the cutoff frequency condition (

http://en.wikipedia.org/wiki/Cutoff_frequency ) has to be included in (

http://en.wikipedia.org/wiki/Microwave_cavity#Cylindrical_cavity) as an additional condition (which we did not do in previous discussions of mode shapes: some of the mode shapes we discussed for certain geometries do not have Q resonance because they are cutoff ! ).

**Not all values of the longitudinal mode shape quantum number "p"=0,1,2,3,... are allowed. Low values of "p" (starting with p=0) are disallowed by the cutoff condition for small diameters. **The cutoff frequency condition (

http://en.wikipedia.org/wiki/Cutoff_frequency ) is that the frequency of the cavity has to be greater than the speed of light in the medium times the appropriate zero Bessel function , divided by Pi*Diameter:

fmnp > cmedium Xfunction / (Diameter * Pi)

where

cmedium = c / Sqrt[epsilonr * mur] where epsilonr is the relative electric permittivity (dielectric constant), and mur is the relative magnetic permeability, and c is the speed of light in vacuum

and where

Xfunction=X

_{mn} for TM modes and Xfunction = X'

_{mn} for TE modes

Now, for a cylinder it is clear what "diameter" means, since the diameter is constant along the length. However, what is the cutoff condition for a truncated cone? This is included in the exact solution I obtained for a truncated cone and it is simply dictated by the small diameter of the truncated cone (which one would expect based on physical reasons):

fmnp > cmedium Xfunction / (Dsmall Pi)

where Xfunction=X

_{mn} for TM modes and Xfunction = X'

_{mn} for TE modes

This has very important consequences for a truncated cone, because as the small Diameter approaches zero, the cutoff frequency approaches infinity. This means that

** for a pointy cone, there can be no resonant Q solution possible. A pointy cone can only have evanescent waves** (since only real solutions to the eigenvalue equation are standing-wave resonant Q solutions, and the complex value solutions to the eigenvalue equation are evanescent waves).

EDIT:

This can also be stated as a cutoff wavelength, where the wavelength lambda

_{mnp} is defined as the ratio of the speed of light in the medium, to the frequency f

_{mnp}:

lambda

_{mnp} = cmedium / f

_{mnp}substituting this, we obtain the following condition

lambda

_{mnp} < (Dsmall * Pi) / Xfunction

**the wavelength for a given mode mnp needs to be smaller than the smallDiameter*Pi /Xfunction**longer wavelengths get cutoff

Another very important consequence is that the cutoff frequency:

fmnp > cmedium Xfunction / (Dsmall Pi)

**implies a minimum diameter for the truncated cone to support a given mode shape at a given frequency:**

Dsmall > cmedium Xfunction / ( fmnp Pi) where Xfunction=X

_{mn} for TM modes and Xfunction = X'

_{mn} for TE modes

**For example, if one wants resonance at fmnp = 2.45 GHz, as @aero was proposing, then it immediately follows that the minimum diameter has to be****Dsmall > cmedium Xfunction / ( 2.45 GHz) **Since the Xfunction increases with m and n (except for one particular X' value for the TE mode with m=1: X'

_{1n} or TE

_{1n}) , small diameters cut off frequencies such that the mn modes can only occur at higher p values, that's why @aero found that the longitudinal mode shape quantum number p=3 for the NASA truncated cone with the dielectric, instead of p=0 or p=1 or p=2.

Dsmall > cmedium Xfunction / ( fmnp Pi)

or, since cmedium = c / Sqrt[epsilonr * mur]

Dsmall > c * Xfunction / (Sqrt[epsilonr * mur] * fmnp * Pi)

**The higher the value of the dielectric constant epsilonr (everything else held constant), the lower the Diameter can be for a given frequency**The higher the value of the dielectric constant epsilonr (everything else held constant), the lower the frequency will be for a given mode shape mnp.

This clearly shows the function of the dielectric in the EM Drive: the dielectric allows mode shapes to occur at frequencies that would otherwise be cutoff for a given geometry.The dielectric allows mode shapes to occur for smaller values of the small diameter.

Also, the cutoff frequency condition clearly shows why NASA and Shawyer located the dielectric at the small end of the cavity, since the cutoff frequency is a function of the the dielectric constant, such that lower dielectric constant allows for lower cutoff frequencies or equivalently smaller diameters.

** The best place to put the dielectric is at the small diameter in order to allow mode shapes to occur that would otherwise be precluded because of the cutoff frequency condition.**

It is very interesting that there is a "sweet spot" of geometries for these EM Drives: too large a diameter leads to too many mode shapes very close to each other (and hence very difficult to tune and keep the EM Drive at a given resonant frequency) while too small diameters lead to cutoff of mode shapes.

** A pointy cone cannot have Q resonant modes, a pointy cone can only have evanescent waves.** The closest one can come up to a pointy cone Q resonance would be to calculate and design a geometry with a dielectric located at the small end and a small but finite Dsmall diameter. Dsmall must be greater than zero, and for practical solutions this implies a minimum diameter.

I will write more on this and give numerical examples and mode shapes for the NASA and the Shawyer experiments. It is all falling into place very nicely now