Evolution equations for the oscillations in a cavity filled with a dynamic medium

Development of the time-harmonic electromagnetic theory dates back to the end of 19th century stating that all the electromagnetic field quantities vary harmonically in time. This assumption, being formalised in the system of Maxwell´s equations (ME) with time derivative, changes the hyperbolic type of the partial differential equation into the elliptic type. Therefore, the time-harmonic solutions to the ME of elliptic type do not satisfy the causality principle and have difficulties with the relativistic transformations. Progress of the time-domain field theory has started via the development of computational electromagnetics in the second part of the 20th century. Numerical techniques are capable of providing huge volumes of numerical data; however, a deep physical insight to the data can be achieved under the presence of analytical results. Nevertheless, analytical theory of time-domain electromagnetics goes through an initial step of its development.