EmDrive Cavity modes The EMDrive described in patent application GB 2493361 is a simple section of a sphere surrounded by a superconducting cooling system.  I have attempted to solve the oscillation modes possible and find spherical Bessel functions for radial directions and Legendre polynomials for "theta".  When I apply boundary conditions, $E_r$ must be zero at the side walls so $P_n^0$(cos($\theta_w$)) = 0 can only happen for specific angles.  If the wall is not at the right angle, $E_r$ must be zero every where.  Same is true for $E_\phi$, but that has zeros for $P_n^1$(cos($\theta_w$)) so it would be a different mode.  
The claim in the patent is that putting RF into the cavity creates a force - but it does not say in what direction!  The EmDrive web site talks about group velocity, but I don't see how that makes sense for a standing wave in a cavity.  What am I missing?  I can find $B$ from $\nabla\times E$), but is there an easy way to find energy density directly from the $E$ field solutions?  Once I know energy density, finding forces should be straight forward.
What I expect is the net total force to be zero, but it would be nice to prove it.
 A: So there is not an easy way - I had to grind out all the algebra.  Starting with $$\nabla^2 E +k^2E = 0$$ The Electric field forms are $$E_r = P_n^0(cos\theta)[\frac{A_n j_n(k r)}{k r} + \frac {B_n y_n(k r)}{k r}]$$ $$E_\phi = P_n^1(cos\theta)[C_n j_n(k r) + D_n y_n(k r)]$$ Using $\nabla \cdot E = 0$ and the symmetry over $\phi$ I find $$E_\theta = -\frac {P_n^1(cos\theta)} {n(n+1)} \{ A_n [\frac {j_n(k r)}{k r} + j'_n(k r)] + B_n[ \frac {y_n(k r)}{k r} + y'_n(k r)]\}$$  Clearly $n = 0$ is not allowed.
Using $\nabla \times E = i \omega B$ I find the following for the $B$ field $$B_r = \frac {-i} \omega k \frac {P_n^0(\cos\theta)} {n(n+1)}[C_n \frac {j_n(k r)} {k r} + D_n \frac {y_n(k r)} {k r}]$$ $$B_\theta = i \frac k \omega  P_n^1(cos\theta) \{C_n  [\frac {j_n(k r)}{k r} + j'_n(k r)] + D_n [\frac {y_n(k r)}{k r} + y'_n(k r)] \}$$ $$B_\phi = \frac {-i} \omega k \frac {P_n^1(cos\theta)} {n(n+1)}[A_n j_n(k r) + B_n y_n(k r)]$$  Note that the imaginary factor simply means that the B field is 90 degrees out of phase with the E field in time.
Now let's look at boundary conditions.  Both $E_\phi$ and $E_\theta$ are zero at $r_1$ and $r_2$. We also have $E_\phi$ and $E_r$ are zero at $\theta_w$. The actual field is the sum over all $n$ but each mode is orthogonal to all other modes so each mode must match the boundary conditions independently.  For $E_\phi$, given arbitrary $\theta_w$, $P_n^1(cos \theta_w) \neq 0$ so all coefficients $C_n$ and $D_n$ must be $0$. That knocks out $E_\phi$, $B_r$ and $B_\theta$.  The same is true for $E_r$ - which says that unless the walls are at a specific angle which is a zero for $P_n^0$ or $P_n^1$, no fields will resonate.  
In other words, if the EmDrive guys don't build the cavity to specific angles, it will simply reflect all power and won't have any RF in it at all!  So let's suppose they DO build it to a specific angle so it will resonate, then the Poynting vector is $S = E \times B$.  Only $E_\theta B_\phi$ is in the $\hat r$ direction, but this is $0$ at the $r_1$ and $r_2$ walls.  So even if it resonates, it won't push. QED.
Life gets much more interesting if there is a center pipe so we have $\theta_1$ and $\theta_2$.  Then we have both $P_n^m$ and $Q_n^m$ as angular solutions.  It will be possible to find a set of coefficients which match all boundary conditions and the system will resonate.  (Replace all $P_n^m$ above with ($P_n^m(cos \theta) + K_n Q_n^m(cos \theta))$
So that's a rather terse reply that covers some 20+ pages of algebra (and calculus I suppose).  It is obvious on its face that the EmDrive can't work, but it can't work in so many ways it's ridiculous.  The resonance of a spherical cone is still an interesting problem and hopefully someone will find this useful.
