Scattering of torsional waves in circular rods from diameter discontinuities

We have studied the propagation and scattering of torsional waves in circular rods with a free surface where the rod diameter is subject to abrupt diameter changes. We applied a mode expansion technique involving propagating and cut-off modes to obtain numerical results for a range of parameter values in the single-step case. The calculated results were compared with simplified models valid in the high and low frequency regions, respectively. We discuss how the results depend on the number of modes used in the expansion. The calculations show that higher radial modes have strong influence at higher frequencies. In particular, near the cut-off frequencies both amplitude and phase responses are strongly influenced. We have found that in order to obtain an accuracy that is sufficient for typical use, all propagating modes plus a few cut-off modes should be included in the calculations. We have also calculated the transmission for a two-step device constructed in analogy with quarter-wavelength transformers well known from bulk wave propagation. Simulations show that this device is expected to perform well for low frequencies, whereas the response deteriorates for higher frequencies when several propagating modes occur in the device