Periodical Focusing Phenomenon of Ultrasonic Guided Waves in Pipes

We put forward a focusing formula to describe the guided waves periodical focusing phenomenon, which, apart from being a fundamental problem in the guided waves' propagation in the pipes, has essential applications in the field of nondestructive evaluation. Due to the partial circumferential loads or nonaxisymmetric defects, the guided waves are not only in the zeroth-order axisymmetric forms but also in the higher order nonaxisymmetric forms. When multiple orders of the same mode exist simultaneously, the angular profile is adopted to describe the circumferential energy distribution of the superposed wave field. However, the angular profile varies in the propagation process. In our finding, this variation is periodic, meaning that the circumferential energy will repeat the process of dispersing and focusing. Thus, we put forward a focusing formula to describe the phenomenon. The proposed formula indicates that the angular profile varies periodically with the ratio of propagation distance to wavenumber, and this period is only related to the pipe radius. Thus, three factors, including propagation distance, excitation frequency, and pipe radius, will affect the angular profile. Moreover, we established an experimental system to verify this phenomenon, based on which we have designed three groups of experiments to investigate these three factors. The experimental results are in good agreement with the theoretical predictions.