We present here the solution of the eigenvalue problems for the open metamaterial square and circular rod waveguides. The Maxwell's equations for the electrodynamical analsis of the open waveguides were solved by the Singular Integral Equations' (SIE) method and partial area method. Our SIE method is pretty universal and let us rigorously analyze open waveguides electrodynamically with any arbitrary cross-sections taking into account of the edge condition. The false roots did not occur applying the SIE method. The waveguide media can be of strongly lossy materials. The signs of the complex permittivity and permeability can be positive or negative in different combinations. We used our computer algorithms based on the two mentioned methods with 3D graphical visualization in the MATLAB language. We present here our numerical calculations of the metamaterial square waveguide with sides equal to 5×10-3m and the metamaterial circular waveguide with the diameter equal to 5×10-3m. We present dependences of phase constant and attenuation constant of metamaterial waveguides at the frequency range from 75 GHz till 115 GHz. We have compared the three dimension (3D) electric field distributions of the main mode and the first higher mode propagating in the square and circular metamaterial waveguides. The calculations of the electric fields were fulfilled at approximately 10000 points in every cross-section. We discovered that the electric field is concentrated at the waveguide boundary. The distribution of the electric field along the perimeter of the waveguide is not uniform. There are two areas on the perimeter of the square and circular waveguides where the electric field has maximum values. These areas are shifted relative to each other on π radians.
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