Torsional rod waves in tapered regions

Previously we have studied the scattering of torsional waves in circular rods with a free surface when the rod diameter is subject to abrupt diameter changes. Here we discuss the propagation of torsional waves in tapered regions and in particular, transmission of the fundamental torsional mode. The calculations are based on a mode expansion technique where propagating as well as cut-off modes are involved. The tapered region is approximated by several steps separated by uniform rod parts arranged so that they closely render the taper shape. Transmission matrices for all steps and uniform parts are combined to yield the desired transmission coefficient through the taper region. Numerical results of the frequency variation of fundamental mode transmission are presented, depending on geometrical parameters such as taper length and the diameters of the two uniform adjacent rod parts. The effect of taper shape on the transmission is demonstrated