Theoretical analysis of the dispersion of Lamb waves forming a wave packet of finite-bandwidth using the method of multiple scales

Abstract

In ultrasonic guided wave inspections, incident waves propagate as a finite-bandwidth wave packet. During propagation, the shape of the wave packet is distorted because of dispersion even if there are no viscous or scattering effects. Although the presence of distortion is well known, how the wave packet is distorted has not been sufficiently investigated. We use the method of multiple scales (MMS), a perturbative scheme, to investigate the dispersion of the Lamb waves comprising the finite-bandwidth wave packet. Solutions obtained by the MMS to order O(ε2) show that the shape of an input wave-packet remains invariant during propagation along the waveguide. The wave packet propagates with a group velocity obtained from a classical analysis. The O(ε3)-order MMS solutions exhibit a symmetric distortion of the shape, whereas the O(ε4)-order MMS solutions feature symmetric and anti-symmetric distortion. For the derivation of the O(ε3)- and O(ε4)-order MMS solutions, the Fourier transform of the wave-packet shape is introduced. The proposed MMS solutions were verified by comparing them with the numerical time-domain solutions constructed from the classical frequency-domain solutions.The strength of the distortion depends on propagation mode, input frequency, and shape of the input wave packet. Because the wave-packet distortion induces a shift and reduction of the peak, the propagation velocity is different from the group velocity. For long-distance inspections, the proposed method provides the precise propagation velocity and details of the propagation-mode selection.