Testing of generalized Helmholtz equations for gyrotropic waveguides

Universal generalized Helmholtz equations were tested for gyrotropic waveguides with arbitrary orthogonal cross-sectional shapes with arbitrary magnetization and general Helmholtz equations for gyrotropic waveguides with arbitrary orthogonal cross-sectional shapes with longitudinal magnetization, which are obtained from generalized Helmholtz equations. From the general Helmholtz equations for gyrotropic waveguides are derived the partial Helmholtz equations for gyrotropic waveguides with certain orthogonal cross-sectional shapes (in particular: rectangular, round, elliptical) with a specific (longitudinal, normal, tangent) magnetization. General Helmholtz equations for gyrotropic waveguides with arbitrary orthogonal cross-sectional shapes in longitudinal magnetization and partial Helmholtz equations for gyrotropic elliptic waveguides in longitudinal magnetization are used during testing.


Introduction
In [1,2], universal generalized Helmholtz equations were obtained for gyrotropic waveguides with arbitrary orthogonal cross-sectional shapes under arbitrary magnetization, which allow to derive common Helmholtz equations for gyrotropic waveguides with arbitrary orthogonal cross-sectional shapes under specific (longitudinal, normal and tangent) magnetization. In [2], the general Helmholtz equations for gyrotropic waveguides with arbitrary orthogonal cross-sectional shapes in longitudinal magnetization were obtained from the generalized Helmholtz equations, from which the partial Helmholtz equations for gyrotropic elliptic waveguides in longitudinal magnetization were then derived.
For the first time, the generalized Helmholtz equations obtained in [2] for gyrotropic waveguides with arbitrary orthogonal cross-section shapes under arbitrary magnetization and the general Helmholtz equations for gyrotropic waveguides with arbitrary orthogonal cross-section shapes under longitudinal magnetization, as well as the partial Helmholtz equations for gyrotropic elliptic waveguides under longitudinal magnetization, need careful testing.
Thus, the purpose of this work is to test the general and generalized Helmholtz equations for gyrotropic waveguides, as well as the partial Helmholtz equations for gyrotropic elliptic waveguides during longitudinal magnetization.

Testing of general and generalized Helmholtz equations
In [2], when determining the partial Helmholtz equations for gyrotropic waveguides with different forms of orthogonal cross-sections for a particular magnetization, the following equations are successively calculated: 1) Generalized equations, 2) General equations, 3) Partial equations.
Therefore, a comparison of the partial Helmholtz equations defined according to the scheme mentioned above with the well-known partial Helmholtz equations from the literature allows us to test the entire chain of calculations.  x In [2], from the generalized Helmholtz equations (1) and (2), the general Helmholtz equations of HEand EH-waves were obtained for gyrotropic waveguides with arbitrary orthogonal cross-sectional shapes with longitudinal magnetization, for the HE-wave components of the ferrite magnetic permeability tensor [6 -8].
Further, from the general Helmholtz equations (4) and (5) we derive the particular Helmholtz equations of HE-and EH-waves for gyrotropic rectangular and circular waveguides with longitudinal magnetization. To pass to specific shapes of the cross-section of the waveguide, we use the formula for the relationship between the Lamé coefficients and the basis vectors 3 where z y x , , -cartesian coordinates; i x -orthogonal curvilinear coordinates (i=1, 2, 3). Therefore, the Christoffel symbols of the second kind in curvilinear orthogonal coordinate systems, expressed in terms of the Lamé coefficients, will be [3]  To derive the particular Helmholtz equations for a gyrotropic rectangular waveguide with longitudinal magnetization, from (6) and (7) we obtain the corresponding Lamé coefficients and Christoffel symbols for the Cartesian coordinate system   Then differential operators of the second order (3) taking into account (8) (9) Substituting (9) into the general Helmholtz equation (4), we obtain the well-known particular Helmholtz equation of the HE -wave for a gyrotropic rectangular waveguide with longitudinal magnetization [5] . 0 Further, substituting (9) into the general Helmholtz equation (5), we obtain the well-known particular Helmholtz equation ЕН -wave for a gyrotropic rectangular waveguide with longitudinal magnetization [5] To derive the particular Helmholtz equation for a gyrotropic circular waveguide with longitudinal magnetization, from (6) and (7) (4), we obtain the well-known particular Helmholtz equation of the HE -wave for a gyrotropic circular waveguide with longitudinal magnetization [5] . 0 Substituting (11) into the general Helmholtz equation (5), we obtain the well-known particular Helmholtz equation ЕН -wave for a gyrotropic circular waveguide with longitudinal magnetization [5]   0 1 1 The testing has shown the correctness of both the general Helmholtz equations of HE-wave (4) and EH-wave (5) for gyrotropic waveguides with arbitrary orthogonal cross-sectional shapes with longitudinal magnetization, and the generalized Helmholtz equations of HE-wave (1) and EH-wave (2) for gyrotropic waveguides with arbitrary orthogonal cross-sectional shapes with arbitrary magnetization.

3.Testing the partial Helmholtz equations
Testing of the particular Helmholtz equations for a gyrotropic elliptic waveguide with longitudinal magnetization is carried out according to the following scheme:  Derivation of the well-known particular Helmholtz equations for a gyrotropic circular waveguide with longitudinal magnetization from similar Helmholtz equations for a gyrotropic elliptic waveguide with longitudinal magnetization, obtained from the general and generalized Helmholtz equations [2], by limiting changes in the shape of the waveguide cross-section from an ellipse to a circle.
 Comparison of the derived particular Helmholtz equations for a gyrotropic circular waveguide with longitudinal magnetization with the known analogous equations from [5]. Figure 1 shows families of confocal ellipses and hyperbolas that form an elliptic coordinate system [9]. The families of these curves intersect orthogonally and the intersection points, in the Cartesian system, have coordinates [9,10]  The angle φ takes values from 0 to 2π. If there is a stretched elastic membrane fixed between two similar ellipses, then the angle ξ changes between the values corresponding to these ellipses.  [9] it follows that if the value of the major semiaxis s is constant and the eccentricity E → 0, then the value of the angle    and the ellipse tends to a circle with radius r = s. To derive the Helmholtz equations for a gyrotropic circular waveguide with longitudinal magnetization from analogous Helmholtz equations for a gyrotropic elliptical waveguide by changing the shape of the cross-section to the limit from an ellipse to a circle, we turn to [9]. Figure 2 shows the differentials of the arcs of the ellipse and hyperbola. According to [9], the differentials of the arcs of the ellipse and hyperbola are defined The partial derivatives included in (12) are equal to [9]  , enclosed between two adjacent pairs of intersecting confocal ellipses and hyperbolas [9] Then, taking into account (13), expression (12) takes the form Then, according to [5], for е = const and ξ → ∞ from (14) and (15) where r -circle radius.