Edge resonance in discrete elastic waveguides and its convergence to isotropic continuous media

We discuss the problem of edge resonance for Lamb waves in a semi-infinite discrete elastic strip, represented by a triangular lattice. In analogy with the reflection problem in the corresponding continuum, for real frequencies the edge resonance phenomenon for the lattice strip is characterised by localised vibrations at its free edge. By characterising the reflection coefficient in the complex frequency plane, we then verify the existence of a complex edge resonance frequency for the lattice, associated with a mode of the homogeneous problem without incident wave. Importantly, when the number of rows in the strip of fixed width is large, we show that the lattice edge resonance frequency converges to the corresponding frequency in the analogous continuum problem for the effective strip. Interestingly, convergence to the complex edge resonance frequency is monotonic only with respect to its real part, while its imaginary part exhibits a minimum absolute value, numerically undistinguishable from zero, for a lattice strip with 65 rows in the transverse direction.