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The Fields Expressed as Two Index Sums 51
Resonance with Cavity Modes
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_(TM accelerator applied voltage approximation azimuthal azimuthal distribution Bessel functions Bevatron bottom surfaces boundary condition circulating current circulation frequency considered constant cosh curve density distribution of particles effect electric field electromagnetic fields electromagnetic modes energy loss evaluated expression fn(ps Fourier coefficients Fourier components FREQUENCY MODULATION given by Eq Graph Hamiltonian harmonic induced voltage inserted integral joule heating machine magnetic fields Maxwell's equations n(po number of particles numerical obtain order of magnitude parameters particle accelerator particle circulation particle distribution phase velocity Poynting vector radiation ratio resonance is possible result rf cavity scalar potential shunt impedance solution space-charge stable-phase synchronous particle synchrotron phase space tangential component TM modes top and bottom toroidal cavity total number total voltage transverse vacuum chamber vacuum tank valid values variable vector potential Vlasov equation zero