The paper is devoted to the theory of eigen electromagnetic waves propagating across the axis of symmetry in waveguides with a non-circular cross-section. The case of waveguides filled with isotropic cold plasma is studied theoretically. Plasma particles motion is described in fluid approximation; expressions for the waves' fields are derived from Maxwell equations. Cross-section of the studied waveguide is modeled by Fourier series with coefficients, which values are less than unity. This allows one to apply method of successive approximations for analytical research of this problem. Boundary conditions, which are formulated in non-linear form over the small parameters of the problem have been applied for derivation the dispersion equations, which determine frequency spectrum of these surface waves for waveguides of different constructions. Studied eigen electromagnetic waves propagate in the form of wave packets, which are approximately described by the main azimuthal harmonic and two nearest satellite spatial harmonics. Represented results have been obtained both analytically and numerically. Possible spheres of the studied eigen waves are discussed.
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