Study of a time variant cavity system

A cavity with a time signal applied taken jointly is treated as a time variant physical system. Standard formulation of the boundary-value problem for system of Maxwell's equations (with partial derivative(t)) is supplemented with the initial conditions and the causality principle. It is proved that electromagnetic field can be presented as the classical decompositions in terms of the solenoidal and irrotational modes but with time dependent modal amplitudes. For the latters, evolutionary (i.e., with time derivative) ordinary differential equations are derived and solved analytically, in quadratures. Simple explicit solutions (in elementary functions) for the modal amplitudes as functions of time are obtained in a particular case. Numerical examples are exhibited, time-domain resonances are studied.