Local Peaks Search Method for Solving Lamb Waves’ Dispersion Equation of Laminated Structures and the Application

Abstract

To study the acoustic characteristics of sound scattered from laminated structures such as elastic plates and shells, it is usually required to solve the Lamb waves’ dispersion equations. Many traditional root-finding methods such as bisection, the Newton–Raphson method, and the Muller method are not able to tackle the problem completely. A simple but powerful method named local peaks search (LPS) is proposed to overcome their drawbacks. Firstly, the non-zero part of the dispersion equation is defined as the dispersion function, and its reciprocal is used to transform the zeros (i.e., roots) into local peaks. Secondly, the chosen complex domain is discretized, and the coarse local domains where the local peaks exist are determined by the direct search method globally. Thirdly, the Muller method is applied to obtain the refined locations of local peaks. Lastly, in order to refine the results, a hierarchical scheme is designed and the iteration of the above procedures is implemented; the error is set to be 10−16 as the stop criteria. The accuracy of the LPS method is validated by comparing it with the bisection method for the problem of elastic plates in the vacuum. The acoustic echo structures are analyzed experimentally. By computation of Lamb waves’ phase velocity, the critical angles are derived numerically and compared with the results acquired by an experiment using monostatic sound transducers. In this way, it is validated that the elastic scattered wave components are the highlights shown in the time-angle figure. Furthermore, the work can be applied for non-destructive testing, especially underwater structural health monitoring.