Lamb wave propagation in inhomogeneous elastic waveguides

The problem of Lamb wave propagation in an axially multi–layered waveguide is treated by a multi–modal approach. A general formalism is proposed that avoids the numerical divergence due to evanescent modes and that is based on an impedance matrix. To describe the fields, we choose a 4–vector composed of the displacements and the horizontal stresses. Due to symmetry properties of the right– and left–going modes, this 4–vector can be split into two 2–vectors described by only two sets of modal components. Moreover, the modal 2–vectors have a biorthogonality relation that allows us to express the fields continuity at the interface between two media in a simple manner. Formally, this approach permits us to extend the multi–modal formalism from fluidic to elastic waveguides. In this context, the impedance matrix is defined as the linear operator that links the two sets of modal components. As in the fluidic case, the impedance matrix has the advantage of avoiding numerical divergence, and can be used to obtain the reflection and transmission matrices, as well as the wave fields. The technique is validated in the case of two semi–infinite elastic plates bounded along their lateral faces (succession of two media) and is also applied to a thick bonding (succession of three media) and to a periodic waveguide (succession of multiple media).