Approximation of the Field Components of TE and TM Modes in Waveguides and Cavities with General Cross Section
This paper shows that the field components of TE and TM modes of waveguides and cavities with general cross section can be approximated by a power series consisting of a separeble solution and nonseparable solutions of Helmholtz wave equation corresponding to the rectangular waveguide.
Use of nonseparable solutions of Helmholtz wave equation in waveguides and cavities
The letter shows how some recently discovered nonseparable solutions of the Helmholtz wave equation can be used in the synthesis of waveguides and cavities with nonconventional cross-sections, and also investigates some interesting properties.
Non-separable solutions of the Helmholtz wave equation
A set of solutions not obtainable by the method of separation of variables is presented for the vector Helmholtz wave equation in circular cylindrical coordinates limited to non-angular dependence. These are constructed of Bessel and trigonometric functions. For example, if A is the vector, the r-component of the simplest member of the set is A, = Ci[mrJ0(pr) cos (mz) + pzJ^pr) sin (mz)]e'"', where Ci is an arbitrary constant, m and p are propagation constants, and co is angular frequency. Brief reference is made to three-dimensional solutions in rectangular coordinates. Introduction. Vector and scalar wave equations appear in electrodynamics, in elastodynamics, and in many other problems involving radiation of energy. During an attempt to formulate, in closed form, the natural modes of elastic vibration of an isotropic solid cylinder of traction-free surfaces and finite dimensions, the writer uncovered some non-separable, simple harmonic solutions that are not described in the standard works [1-8] on partial and ordinary differential equations. This brief paper is a presentation of these solutions in circular cylindrical coordinates, as befit the geometry of the elastic problem, and the paper concludes with some remarks about non-separable solutions in rectangular coordinates. In vector form the Helmholtz wave equation is [£ + k2] A = 0, fc2 = (co/c)2 (1) where & is the vector Laplacian operator,1 co is 2x times the frequency, c is the velocity with which a constant phase is propagated, and A is a field vector. Equation (1) represents three scalar partial differential equations customarily solved by the method of separating variables. In his elastic problem the writer assumed axial symmetry to avoid dependence upon the angular coordinate. With this simplification the separable-type solutions of Eq. (1) are P) 2Vi(pr)J (sin mz. / J oCp^l/si: \iV0(pr)Ac< sin mz cos mz 'e~ i (3) *Received December 12, 1963. This work was supported by the Air Force Office of Scientific Research under Contract AF 49(638)-1148. 'This notation has been introduced in [9].