Anomalous dispersion of guided wave in cylindrical multi-layered solid media

This paper presents an investigation into the dispersion of guided waves in cylindrical multi-layered solid media. This will aid in the design of ultrasonic transmission rods, consisting of a metal rod surrounded by a sheath, used in liquid metal cleanliness measurements. The aim is to understand the material parameters which define the regions of normal and anomalous dispersion, such that the sheath material and transmitted frequency range can be designed to improve measurement of the surrounding medium. Three models were created for infinite length rods: (1) in a vacuum, (2) embedded in an infinite medium, and (3) surrounded by an infinite medium connected via an adhesive intermediary layer (i.e. single-, double and three-layered models). The dispersion equations were transformed into real functions and solved numerically using the bisection technique, to give robust solutions of the phase and group velocities for all propagating modes. Simulations were performed for a variety of adhesive layers and infinite medium material properties. In the single-layered model (1) it was found that only first order modes show the attributes of anomalous dispersion. In the double-layered model (2) the materials were chosen to satisfy the Stoneley wave existence criterion and the rod density was chosen to be greater than the outer medium. In this case only the Stoneley modes were predicted to exhibit signs of anomalous dispersion. These were found to exist in two distinct frequency bands. In the three-layered model (3) the Stoneley mode was found to exhibit anomalous dispersion when the density of the inner rod was larger than that of the adhesive layer, whereas the flexural normal modes required both the shear velocity and density of the rod to be greater than those of the adhesive layer. These results suggest that the high-frequency anomalous dispersion regions could be related to the interface wave properties to aid in transmission rod design.