Helmholtz equation solutions corresponding to multiple roots of the dispersion equation for a waveguide with impedance walls

The Helmholtz equation solutions corresponding to multiple roots of the dispersion equation are considered for a waveguide with impedance walls. It is shown that in this case the eigenfunctions are determined by expressions that cannot be obtained by the separation of variables. In addition to exponential or trigonometric factors, such functions involve factors linear in coordinates. Equations for calculating the multiple roots are obtained, and the relationships that determine the impedance values at which the multiple roots appear are presented. The Green’s function is constructed for the case of the appearance of multiple roots. A plane waveguide and a circular waveguide with axially symmetric modes are considered.