Solution to the Klein–Gordon equation for the study of time-domain waveguide fields and accompanying energetic processes

In electromagnetics, power flow and energy densities are associated with time-varying electromagnetic fields. For time-harmonic waves, such quadratic characteristics are derived using the complex-valued spatial forms of field vectors resulting in time-averaged power flow and energy densities. Hence, the opportunity to study dynamic processes of the quadratic characteristics is lost in the time-harmonic electromagnetics. To overcome this drawback, the authors consider the problem of a waveform propagation in a hollow waveguide via solving the system of Maxwell´s equations with time derivative in an energetic space of `real-valued´ functions. The transverse electric and transverse magnetic modal field components are presented as the products of modal basis elements and the modal amplitudes. The vector basis elements are obtained with required physical dimensions, volt per metre and ampere per metre. Thereby, the modal amplitudes are dimension-free scalar functions. The system of evolutionary equations for the modal amplitudes is derived and solved explicitly. The Klein-Gordon equation (KGE) plays a central role herein. A novel set of real-valued solutions to the KGE is obtained and applied for analysis of the energetic processes pertinent to the time-domain signal propagation. The velocity of transportation of the modal field energy is obtained and energetic exchange between the transverse and longitudinal field components is discussed.