Evolutionary Approach to Solve a Novel Time-Domain Cavity Problem

The problem of forced oscillations in a cavity filled with a dynamic plasma is solved in the time domain in compliance with the principle of causality. Interaction of the plasma and the cavity field is driven by the motion equation where the field is present in the Lorentz force standing at its right-hand side. The solution has a form of the modal expansions. Every term herein is a product of two multipliers. One is the vector element of the modal basis dependent on coordinates only. The basis is obtained as an eigenvector set of a self-adjoint operator specially selected from Maxwell’s equations. The operator is specified as a merger of the differential procedure
$\nabla \times $
and the boundary conditions over perfectly conducting cavity surface. The basis elements are derived with their physical dimensions, i.e.,
volt per meter
and
ampere per meter.
The other factor is a scalar dimension-free time-dependent modal amplitude. A system of
evolutionary
equations (i.e., with time derivative) for the modal amplitudes is derived and solved explicitly under the initial conditions. The problem is solved in the Hilbert space of
real-valued
functions of coordinates and time.