1) I have re-run the calculations using the Volumetric Mean instead of the Geometric Mean. Please compare the results shown below using the Volumetric Mean with the GeometricMean results previously shown in post

http://forum.nasaspaceflight.com/index.php?topic=36313.msg1319117#msg1319117 .

2) We define the Volumetric Mean as follows:

VolumetricMeanDiameter=Sqrt[(aeroSmallDiameter^2+aeroSmallDiameter*aeroBigDiameter+aeroBigDiameter^2)/3]

DERIVATION OF VOLUMETRIC MEAN

Defining the following symbols for the radii of the truncated cone (frustum of a cone):

r=smallRadius=smallDiameter/2

R=bigRadius=bigDiameter/2

The volume of a truncated cone and the volume of a cylinder are:

Volume of the frustum of a cone =Height*Pi*(r^2+r*R+R^2)/3

Volume of a cylinder= Pi*(EquivalentR^2)*Height

Equating these volumes one arrives at an expression for the Equivalent Radius of a cylinder having the same volume as the volume of the frustum of a cone:

Volume of a cylinder = Volume of the frustum of a cone

Pi*(EquivalentR^2)*Height = Height*Pi*(r^2+r*R+R^2)/3

hence

VolumetricMeanRadius = EquivalentR

= Sqrt[(r^2+r*R+R^2)/3]

= Sqrt[(smallDiameter^2+smallDiameter*bigDiameter+bigDiameter^2)/12]

or,

VolumetricMeanDiameter=Sqrt[(aeroSmallDiameter^2+aeroSmallDiameter*aeroBigDiameter+aeroBigDiameter^2)/3]

3) Let's define as "Aero geometry" the following definition for the NASA Brady et. al. cavity:

Aero Best estimate as of 11/9/2014

http://forum.nasaspaceflight.com/index.php?topic=29276.msg1285896#msg1285896 cavityLength = 0.24173 m

bigDiameter = 0.27246 m

smallDiameter = 0.15875 m

then

aeroGeometricMeanDiameter=Sqrt[aeroBigDiameter*aeroSmallDiameter]

= 0.207974 meter

aeroMeanDiameter =(aeroBigDiameter+aeroSmallDiameter)/2

= 0.215605 meter

aeroVolumetricMeanDiameter=Sqrt[(aeroSmallDiameter^2+aeroSmallDiameter*aeroBigDiameter+aeroBigDiameter^2)/3]

= 0.218089 meter

4) Let's define as "Fornaro geometry" the following definition for the NASA Brady et. al. cavity:

Fornaro estimate

http://forum.nasaspaceflight.com/index.php?topic=36313.msg1302455#msg1302455 cavityLength = 0.332 m

bigDiameter = 0.397 m

smallDiameter = 0.244 m

then

fornaroGeometricMeanDiameter=Sqrt[fornaroBigDiameter*fornaroSmallDiameter]

= 0.311236 meter

fornaroMeanDiameter =(fornaroBigDiameter+fornaroSmallDiameter)/2

= 0.3205 meter

fornaroVolumetricMeanDiameter=Sqrt[(fornaroSmallDiameter^2+fornaroSmallDiameter*fornaroBigDiameter+fornaroBigDiameter^2)/3]

= 0.323529 meter

5) Given the experimentally reported frequencies, the geometrical dimensions and the value of speed of light in air, one can invert the frequency equation (see:

http://en.wikipedia.org/wiki/Microwave_cavity#Cylindrical_cavity ) to obtain X

_{m},

_{n} and X'

_{m} values as a function of constants and the longitudinal mode shape number "p". Let's define the error difference between these X

_{m},

_{n} and X'

_{m},

_{n} values and actual X

_{m},

_{n} and X'

_{m} values as:

error= (value of X

_{m},

_{n} or X'

_{m},

_{n} obtained from frequency eqn.)/ (correct value of X

_{m},

_{n} or X'

_{m},

_{n} ) -1

where X

_{m},

_{n} is used for TM modes and X'

_{m},

_{n} is used for TE modes.

6) Then I obtain the following mode shapes and associated errors:

**BRADY "A"****Fornaro Geometry**Best result: TM311 error= - 0.146%

2nd best: TM014 error= - 3.00%

**Aero Geometry**Best result: TE311 error= - 0.427%

2nd best: TE212 error= + 3.05%

**BRADY "B"****Fornaro Geometry**Best result: TM311 error= + 0.0780%

2nd best: TM014 error= - 1.389%

**Aero Geometry**Best result: TE311 error= - 0.192%

2nd best: TE310 error= + 5.35%

**BRADY "C"****Fornaro Geometry**Best result: TM310 error= - 0.0781%

2nd best: TM012 error= + 1.303%

**Aero Geometry**Best result: TE310 error= + 2.292%

2nd best: TE311 error= - 3.43%

TE011 or TM111 error= + 5.88%

**CONCLUSIONS**1) At the frequencies tested by Brady et.al. , what mode-shape corresponds to a given frequency is very sensitive to the exact geometrical dimensions of the cavity. The reason for this is that there are many natural frequencies very close to each other, each of these frequencies having different mode shapes. Therefore, the above-given Fornaro and Aero guesses of the dimensions of the Brady et.al cavity give very different mode shapes for a given frequency.

2) Using the Volumetric Mean, all cases run by Brady et.al, for the Fornaro dimensions correspond to transverse magnetic (TM) mode shapes, all cases run by Brady et.al, for the Aero dimensions correspond to transverse electric (TE) mode shapes:

**Fornaro **dimensions -----> Transverse

**Magnetic **mode shapes : the magnetic field is in the circumferential direction

**Aero **dimensions -----> Transverse

**Electric** mode shapes: the electric field is in the circumferential direction

EDIT: Actually, using the Aero dimensions, all cases run by Brady et.al, correspond to transverse electric (TE) mode shapes, using either the Geometric Mean or the Volumetric Mean to estimate the equivalent cylindrical diameter. I think that the TE mode shapes are the one that should provide thrust because it is only the TE mode shapes that have the magnetic field directed along the longitudinal direction of the EM Drive. Physically, an axial magnetic field may result in a measured thrust either 1) as an artifact, because the magnetic field can heat the flat ends of the truncated cone and hence produce thermal buckling or 2) as a real means of propulsion, by the magnetic field coupling with the Quantum Vacuum, for example.

3) Using the Volumetric Mean, the field mode shapes for the Fornaro dimensions are similar to the field for the Aero dimensions: mode 311 for Brady cases "a" and "b"" and mode 310 for Brady case "c". The difference between them is that for the Fornaro dimensions the circumferential field is magnetic while for the Aero dimensions the circumferential field is electric. Furthermore, one can state that the field mode shapes

*in the circumferential cross section* for the Fornaro dimensions are similar to the field for the Aero dimensions for ALL Brady cases: they are all 31. The difference between them is

Brady cases "a" and "b'' ---> one half-wave in the longitudinal direction

Mode 311 ----> 3 full-wave patterns around the circumference

1 half-wave pattern across the diameter

1 half-wave pattern along the longitudinal length

______________________________________________

Brady case "c'' ---> constant in the longitudinal direction

Mode 310 ----> 3 full-wave patterns around the circumference

1 half-wave pattern across the diameter

constant along the longitudinal length

4) Using the volumetric mean in all cases, the errors are smaller using the Fornaro estimate of geometry, while using the geometric mean in all cases, the errors are smaller using the Aero estimate of geometry. Conclusion: this may be fortuitous and it may not be something to discriminate between the two estimates of geometry.

5) Using the volumetric mean leads to more stable values of mode shape with variation in frequency than when using the geometric mean: it gives mode shape 31 for all Brady cases.

6)

** The most important conclusion**: this exercise has made me appreciate the true role and value of the dielectric polymer, and why NASA reported it was so important. It is evident that at these frequencies there are so many mode shapes bunched next to each other that:

it would be extremely difficult to predict what mode shape one will get with a given geometry at these frequencies, because small variations in geometry lead to large changes in mode shape. Certainly it would be impossible to predict the mode shape of an empty cavity with the coarse finite element model used by NASA Brady et.al.

(There are no Bessel functions in a finite element model: the solution is approximated with low power

*piecewise *polynomials in each finite element. The finite element solution is a Galerkin solution "in an integral sense" and not an exact partial differential solution "point to point through the domain".)

Hence the function of the dielectric polymer is to force the cavity to function into a preferred mode shape.

For NASA Brady et.al. they used a doughnut-shaped dielectric polymer to try to force the cavity to operate in TE01 mode shape (constant electric field in the circumferential direction). The calculations show that Brady experiment"c" was the closest to be such a mode shape while experiments "a" and "c" were much further apart. Hence experiment "c" resulted in a much larger thrust per PowerInput with the dielectric polymer having a constant electric field in the circumferential direction and the rest of the cavity having a constant electric field in the circumferential direction with a half wave amplitude in the longitudinal direction, resulting in a mode TE012. The polymer dielectric will produce an extra longitudinal half wave in the dielectric polymer, resulting in two half waves in the longitudinal direction of the whole cavity: 1) one half wave within the doughnut-shaped polymer dielectric itself, along the thickness of the doughnut and 2) the other half wave within the longitudinal direction of the rest of the empty cavity that has no dielectric polymer.

7) So, an** optimal design of an EM Drive** would proceed as follows: first decide what mode shape provides the largest thrust per power Input. Then design a dielectric polymer shape that would best force this mode shape. Then model the geometry of the cavity such that the rest of the cavity is also in the same cross-sectional mode shape (to amplify the resonance) and accurately model at what frequency this occurs.