Anyone know what would be the effect if the ends of Emdrive, instead of flat surfaces, were semi spherical ?
The bandwidth of the frustum decreases while increasing the resonance within the narrower range (increased q-factor). In Q-factor dependent theories, this should increase the measured thrust for a given center frequency.
My intuition tells me the orthodoxy here of using the ray-tracing concave at apex and base is misguided. I believe fields vs. photons is the approach most optimally taken.
IMHO Higher Q will be realized by using convex at apex and concave at base reflection plates. Sharp, high-divergence bends result in high impedance to currents, hence loss. I would like to see an optimization of Q vs. dispersion for a resonator. Comsol I know can do this.
Loosely speaking, the way to increase Q is to set your cavity such that resonance occurs only at a single, 'pure' frequency rather than a base frequency + a(n infinite ) series of higher order harmonics. Again loosely speaking, this occurs when your cavity and your resonating fields 'belong' to one chosen coordinate system.
What does that mean?
Well, consider a cube. If your cavity is a cube, you'll have 'pure' resonance solutions in cartesian coordinates (x,y,z). Injecting, say, a single rectangular plane wave can cause a strong resonance if the frequency of that wave is well-chosen. Consider now an cylinder. This cavity will have 'pure' solutions expressed using bessel/hankle functions. If you excite the cavity with such, then you can again get strong resonance. If, however, you attempt to excite the cylindrical cavity with a rectangular plane wave (or vice-versa), then you'll have a mis-match between your wave and your cavity, and the malformed wave is then forced to convert into a form that matches the cavity. In layman's terms, this manifests as your original wave breaking into multiple different waves; spreading your energy out among multiple frequencies and modes. If you want a high Q (single, pure mode) this is bad. Therefore, one should chose a wave that matches the cavity.
The problem with a flat-ended frustum is that the cavity itself mixes coordinate systems. Flat endplates are very cartesian, however the curved wall is actually best thought of as a truncated cone, which belongs in the spherical coordinate domain. As such, the resonant modes in the cavity are never quite pure. A well designed setup can still produce good data, but you're always losing energy to this back and forth conversion game. Using curved endplates, such that the cavity geometry becomes a truncated spherical cone, restores the 'pureness' of the setup; and thus much higher Q factors can be achieved.
Well, at least theoretically. Engineering is pretty much the study of getting something built well enough to work as an optimal solution may not actually exist. It's the study of intelligent trade-offs and compormises. High Q is desirable. It's also more difficult to tune, more difficult to build, and more dependent on the rest of your hardware working near-perfectly. If a high-Q frustum falls out of resonance, it's going to fall fast and hard. Flat end plates are easier to work with; and won't bankrupt the home-builders here who are funding their own experiments out of pocket. (At least not as quickly. Some of the guys here are getting pretty into this: I half expect to see a vacuum chamber pop up here soon!)
For the more mathematically inclined, a good treatment is found here regarding closed form solutions to a truncates spherical cone.
http://gregegan.customer.netspace.net.au/SCIENCE/Cavity/Cavity.html I don't remember seeing this posted before, but I'd be very surprised if it hasn't been; all company considered.
The main takeaway from that page is that the sharp curves of the corners can often be rendered insignificant if one choses a (pure!) resonant mode such that the corners are null regions in your standing waves. Scroll down about 3/4 of the way through that page to the "Current, Heating, and Q factors" section and look at the differences in some of the TM shapes. In this case, I believe the benefit to gentle corners would be vastly outweighed by the loss of Q in some of these more exotic cavity shapes.