In a 1975 paper [8], Heinig established the following three inequalities:where A = p/(p + s − λ) with p, s, λ real numbers satisfying p + s > λ,p > 0;where B = p/(2p + sp − λ −1) with p, s, λ real numbers satisfying 2p +sp > λ, + 1, p > 0;where is a sequence of nonnegative real numbers,and C = p[l + l/(p + s−λ)] with p, s, λ real numbers satisfying s > 0, p ≥ 1, and p +s > λ 0.
This paper presents the results of precise numerical computations and extensive analytical investigations of the angular propagation constants for the various electromagnetic modes which may exist in waveguide bends of rectangular cross section. Although the values of the several independent parameters have been chosen to be commensurate with microwave applications, the results are displayed and discussed in a manner which should be convenient for both the analysis and the design of such components in analogous problems in other and diverse practical situations. Moreover, special consideration is given to the relationship of this work to various efforts in the study of radio wave propagation.
Two important approximations, one for “high” frequencies and the other for gradual bends, are also considered as a part of the present inquiry. These approximations, especially the latter, turn out to be remarkably good, and rather accurate values of the propagation constants can be obtained with their use for wide ranges of parameter values.
The following are typical of the conclusions which can be drawn from this research: (1) cutoff frequencies for modes existing in continuously curved waveguides are curvature dependent; (2) there exist waveguide bends such that circularly polarized waves are undistorted by transmission along the curved section; (3) the angular propagation constants are generally monotonically increasing functions of the operating frequency f (for fixed mean radius) and of the mean radius R (for fixed frequency). Moreover, they vary linearly with f for frequencies far above cutoff and with R for small curvatures.
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