Calculating the full leaky Lamb wave dispersion characteristics

Abstract

Lamb waves are mechanical waves guided in a plate with stress-free boundaries. In practice, the plate is often in contact with a fluid. The propagation of guided waves may change substantially in this case due to interaction with the adjacent medium, e.g., acoustic radiation may lead to leakage of energy into the fluid domain. The resulting plate modes are called leaky Lamb waves, and their attenuation due to leakage is an important parameter for the design of ultrasonic devices that exploit such waves. Modern methods to calculate dispersion characteristics of guided waves, i.e., the wavenumber-frequency relationship, solve the corresponding discretized eigenvalue problem. This kind of model for leaky Lamb waves with the analytically exact fluid interaction leads to a nonlinear eigenvalue problem, which is difficult to solve. We present a change in variables that yields an equivalent polynomial eigenvalue problem. The latter can be rewritten as a linear eigenvalue problem in state space. As a result, conventional numerical eigenvalue solvers can be used to robustly determine the sought wavenumbers. In contrast to traditional root-finding of the characteristic equations, the presented method guarantees to find all solutions and naturally handles complex wavenumbers. Additionally, no initial guesses for the wavenumbers are required.