## Application of the generalized Fourier method to the problem of hollow waveguide with triangular cross section

Published: 05/13/2004
Original: www.laboratory.ru
© J. E. Andrushkevich, V. A. Zhiznevsky
© 2004, EDS–Soft,  http://www.eldys.org,   E-mail: publications@eldys.org

An obvious example of realization of advantages of the generalized Fourier method GFD [1] over the classic one in solving of applied electrodynamics problems is the problem of hollow triangular cross section waveguide (Fig.1), whose envelope is considered to be a perfect conductor, while the inner medium is homogeneous. In the most cases, this model turns out to be satisfactory in practical calculations. If necessary, it can be made more accurate by accounting for the losses in the metal.

Usually the search of the electromagnetic field vectors leads to the Helmholtz equation, since the components of these vectors must satisfy this equation:

 (1)

The spatial problem of wave propagation in such longitudinally-homogeneous structure can be reduced to the two-dimensional Helmholtz equation by means of classical separation of the z–variable, that is, by representing function in question in the following form:

 (2)

The equation for takes the form:

 (3)

Here not only the function is unknown, but the parameter meaning the cross wave number is also unknown. The equation (3) in itself does not have definite solutions from the physical point of view. It is necessary to formulate the boundary value problem. It is known, e.g. from [2], that to determine the Е–wave family for some guiding structure with homogeneous medium and perfect conducting walls we must find the solutions to the boundary problem, containing, besides equation (3), the following condition:

 (4)

where L is being understood as a perfect conducting contour of the cross section of the empty waveguide or, in more complicated cases, the sum total of contours. In our example, as it is seen from the figure, a rectangular isosceles triangle becomes L. Applying the classical Fourier method to this boundary problem, i.e representing the function in question in the form:

 (5)

we can obtain the following common solution for the equation examined:

 (6)

The undetermined constants, contained in this solution, must be determined from boundary conditions, but the system of equations, we obtain this way, does not have non-trivial solutions. Therefore, the solution (6) not satisfy the formulated boundary problem. We can divide the surrounding contour into segments, which obviously leads to the increase in the number of boundary problems to be solved. It can be avoided by using GFM.

If the function in question is represented as:
 (7)

the equation (3) is reduced to the bilinear form:

 (8)

At the next stage of the GFM technique, it is necessary to build the matrix of functions for the bilinear equation which in our case looks as follows:

 (9)

In accordance with the theory of implementation of GFM[1], using this matrix, the following systems of divided equations can be constructed:

 (10)
 (11)
 (12)

The systems presented differ by functions included in their basis, and by number of these functions. By analyzing these systems, we find that only the system (11) can have solutions, which meet the requirement of linear independence of the functions in question by every variable. The solution of system (11) provided that has the following form:

 (13)

This solution contains eight uncertain coefficients and the constants of division which must be determined from the boundary conditions.

The condition along the x–axis having the form leads to the equation:

 (14)

From this it follows that

The condition along the y–axis having the form leads to the equation:

 (15)

from which we assume that .

The condition along the hypotenuse of the triangle having the form leads to the equation:

which can be transformed to the following:

 (16)

Solving the given trigonometric equation we can transform it to identity when the following restrictions on the undetermined constants are met:

 (17)

where k,n,m are nonzero integers.

Under these restrictions, the function takes the form:

 (18)

Here the undefined amplitude constant С appeared from the following notation:

Let’s return to the first formulated problem of determining the E-wave family for the directing structure examined. In accordance with [2], the proper functions meaning the longitudinal component of the electric field for the wave determined by numbers m,n appear as . The proper numbers from the expression (17) correspond to these own functions. The general electromagnetic field for this waveguide can be determined from the dependencies of cross components on and , following from the Maxwell’s equations:

where is the longitudinal wave number, and is the angular frequency of the wave process.

References

1. Andrushkevich J.E. About a generalization of the Fourier method of division of variables. EV & ES.1998. №2. (In Russian).
2. Nikolsky V.V., Nikolskaya T.I. Electrodynamics and wave propagation. M.: Science.1989. (In Russian).