Matching of Fundamental Modes at a Junction of a Cylinder and a Truncated Cone; Application to the Calculation of Some Radiation Impedances

Abstract

The problem of the junction between a cylinder and a truncated cone at frequencies below the first cutoff of the cylinder is investigated, in particular for the case of acute angles. An analytical model of the matching of a cylinder and a truncated cone is derived for the general case of a cone of finite length having a known terminal impedance. When the cone is infinite and the angle is right, the problem is similar to the classical problem of a tube radiating in an infinite baffle. The model is based on a general formulation of the junction of several waveguides at low frequencies (when only the fundamental mode propagates in each guide), and on the assumption that at high frequencies, the radiation impedance of the cylinder is equal to its characteristic impedance. The model has the form of an equivalent circuit, and involves several parameters related to the geometry (the areas of the surfaces defining the matching cavity and the volume of this cavity). In addition, the model requires one supplementary parameter only, i.e., the zero frequency value of the added mass (or length correction), which has to be determined numerically (the Finite Element Method is used). Analytical and numerical results agree very well at low and moderate frequencies, up to the cutoff of the first higher-order mode. For the radiation into an infinite flange, the results improve upon those in a recent publication that were obtained by optimization. The case of obtuse angles is more complicated and is briefly discussed. Finally for the case of infinite cones, the reflection coefficient is compared to that obtained in previous studies.