Integral Representations of a Pulsed Signal in a Waveguide

The problem of pulsed excitation of an acoustic waveguide with a constant cross section is considered. Absorption is ignored. As the most general model of such a waveguide, matrix Klein–Gordon equation is investigated (the waveguide finite-element method). For a waveguide described by the model, several field representations are constructed: in the form of a double integral over \({{\omega }}\) and \(k\), in the form of a sum of integrals over \(k\), and in the form of a sum of integrals over \({{\omega }}\). Riemann surfaces of multivalued complex functions \(k\left( {{\omega }} \right)\) and \({{\omega }}\left( k \right)\), which are implicitly determined by dispersion equation, are introduced. Integration in the field representations is held over the sheets of the Riemann surfaces. By deformation of integration contours, equivalence of the aforementioned representations is proved.