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The problem of excitation of electromagnetic oscillations in a waveguide with metal walls, which has an arbitrary cross-section, is reduced to an infinite set of boundary-value problems for telegraph equations in a quarter of the plane. The values of the longitudinal components of the field or of the lateral components of the magnetic vector (surface currents) on the cross-section of the waveguide can be chosen as the wave sources.It is preliminary shown that the components of the non-harmonic electromagnetic field in the waveguide are expanded into series by two sets of eigen functions of the two-dimensional Laplacian that satisfy the Neumann or Dirichlet boundary conditions. The coefficients of these expansions are the solutions of telegraph equations or derivatives of these solutions. The boundary-value problem for the telegraph equation in a quarter of the plane is considered. It has been established which boundary conditions are sufficient for determining its unique solution. The solvability conditions of the auxiliary over-determined boundary-value problem have been written down. The formulas that give an explicit solution of the telegraph equation in a quarter of the plane in the case of different boundary conditions have been obtained. It is shown how to determine the boundary values of the solutions of the telegraph equations for various types of sources of the electromagnetic field.As an example, the longitudinal components of the field for the high mode of a rectangular waveguide for given pulse sources are determined numerically.
