Focussing bending waves via negative refraction in perforated thin plates

We propose a design of a periodically perforated thin plate leading to lensing effect of bending waves via negative refraction. To achieve this goal, we first analyze the band spectrum of the bi-harmonic operator for an array of freely vibrating square voids using both numerical (finite elements) and asymptotic methods. We then find some point in the reciprocal space where the acoustic dispersion surface displays a convex isofrequency contour shrinking with frequency. We finally demonstrate that a point force generating a bending wave above a finite array of 221 perforations displays an image underneath according to the Snell-Descartes inverted laws.