The complete, closed-form exact solutions for the hollow equilateral triangular waveguide are presented. Both TM and TE mode shapes are determined.Introduction: Hollow waveguides with metallic walls have been used extensively in electromagnetic devices. In these cases the Maxwell's equations reduce to the Helmholtz equation [1]. A good source for the methods and solutions for various cross-sections is given by Ng [2]. Of particular interest are the closed-form exact solutions, which include cross-sections such as parallel plates, circular, annular, elliptic, eccentric circular, and equilateral triangular cross-sections. These exact solutions not only solve a fundamental problem, but also serve as accuracy standards for approximate (series, variational, numerical) methods.This Letter considers the exact solution for the equilateral triangular waveguide. Previous works include Schelkunoff [3], who obtained some exact solutions for both TM and TE modes. Later Polya [4] obtained a particular solution, which corresponds to one of the TM modes. We show that both sources are incomplete, and proceed to give a description of all the mode shapes.Consider the equilateral triangular waveguide, the cross-section of which is enclosed by
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