Elastic waves in a fluid-loaded, semi-infinite axisymmetric rod

In a fluid-loaded, semi-infinite axisymmetric rod, a free shear stress boundary condition on the circular cross-sectional end introduces complicated, nondispersive waves in the solid. They are composed of a pulse wave, which has the same waveform as the transmitted one and travels at speed c1, and different kinds of pulse trains, each of which travels along the rod at the speed of either c1 or square root of 2c2, where c1 and c2 are the propagating speeds of the longitudinal and transversal bulk waves, respectively. Furthermore, one can conclude from the solutions to the boundary conditions that c1 and square root of 2c2 are the only phase speeds of nondispersive waves. Frequency equations associated with these waves are established, and the solutions are solved and discussed analytically and numerically. The acoustic field in the fluid is also fully discussed, and it is more complicated than a single outgoing Hankel function as described for an infinite rod. The acoustic energy coupling between the solid and the fluid and the end reflection and transmission are quantified as well. In the end, experimental examinations of the echo spectra, using an aluminum rod immersed in the water and air, fully confirm the numerical solutions to the frequency equations.