Torsional rod wave scattering from tapering between regions of different cross-sections

Previously (Engan 1998) we have studied the scattering of torsional waves in circular rods with a free surface where the rod diameter is subject to abrupt diameter changes. Here, we discuss the scattering of torsional waves from tapered regions and, in particular, transmission of the fundamental torsional mode. The calculations are based on a mode expansion technique in which propagating as well as cut-off modes are taken into account. The tapered region is approximated by numerous steps separated by uniform rod parts arranged so that they closely render the taper shape. Transfer 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 dependence of the transmission on taper shape is demonstrated by calculating the response of various shapes.