Longitudinal stress wave propagation in non-uniform bars

The problem of propagation of a longitudinal strain pulse inan elastic bar of finite length, and the influence of discontinuitiesin the cross- section of the bar on the form of such a pulse, hasbeen investigated both theoretically and experimentally.In the theoretical work, a system of approximate, one-dimensional equations is derived for axially symmetric motions ofan elastic bar of circular cross- section. The equations take intoaccount the coupling between longitudinal, axial shear, and radialmodes. It is found that this theory has three modes of wavetransmission and the lower gives a good approximation for thecorresponding mode oft he exact (Pochhammer) theory. It is alsofound from the group velocity analysis that, according to the presenttheory, the initial disturbance to arrive at a particular station alongthe bar is composed of wave groups of very short wave-lengthconnected with the higher modes (i. e. the radial and axial shearmodes) and travelling with the dilatational velocity Cl’A numerical method, based on the finite-difference technique,has been developed for solving the approximate equations ofmotion and applied for calculating the axial strain in a uniformcylip.drical bar of finite length, subject to a uniform time-dependentaxial pressure at one end while the other end is free of stresses.Colliding bar experiments, exploying cylindrical steel bars withalmost flat ends, have been carried out to correlate with the