Ultrasonic transmission of longitudinal modes in miniaturized cylindrical waveguides

This study investigates the transmission and reflection of axisymmetric longitudinal L(0,n) modes in miniaturized cylindrical waveguides. The theoretical framework developed herein focuses on the analysis of the propagation of longitudinal modes within a homogeneous, elastic, and isotropic cylindrical waveguide with a varying cross-section. The novelty of this paper is found in the energy calculations and the consideration of higher-order modes for accurately describing the transmission and reflection coefficients at waveguide transitions. A 2D axisymmetric Finite Element Method (FEM) is used to calculate these coefficients based on the modal energy of propagating modes across different frequencies. The influence of geometrical and material properties, along with modal density, is examined numerically. Results demonstrate that optimal transmission occurs when the wavenumber ratio of propagating modes in connected waveguides is an integer, leading to effective coupling. Experimental validation on steel rods with diameters from 4 mm to 0.8 mm shows strong agreement with numerical results. It is observed that the L(0,1) mode around 1 MHz is the most suitable mode for efficient transmission between the waveguides. In contrast, the higher-order modes (L(0,2) around 1.25 MHz, L(0,3) between 1.3 and 1.5 MHz, and L(0,4) from 2.25 MHz) exhibit weak or irregular transmission, with more pronounced reflection behaviors, indicating that they are not optimal for efficient transmission in this configuration. These findings underline the importance of an optimized geometric transition in enhancing the transmission efficiency in such miniaturized waveguides.