A harmonic-oscillator analogy for elastic wave localization in piezoelectric metamaterial beams

Locally resonant piezoelectric metamaterials can localize elastic energy at defects, but conventional defects rely on abrupt changes in the lattice that trap the field in isolated cavities and impose sharp spatial constraints. Here we replace the abrupt defect with a smooth piecewise-quadratic modulation of the electrical resonance across an array of inductively shunted unit cells, forming a distributed trapping region whose location, width, and depth are set electrically rather than by geometry. This graded profile maps the slowly varying wave envelope onto the quantum harmonic oscillator, so the trapped modes are Hermite–Gauss envelopes whose spatial localization length and descending frequency sequence are predicted in closed form from the modulation depth and width alone. A coupled electromechanical eigenvalue analysis, an inverse participation ratio, and transient simulations establish the trap. A resonance maximum traps a Gaussian fundamental and a family of higher modes while a minimum expels the field, the depth and the width act as independent controls over the confinement and the spread, and the localized mode relocates anywhere along the beam without altering the structure. Against an abrupt defect, the graded well confines monotonically and isolates the trapped energy, and the confinement is read directly as a voltage through the shunt. A three-dimensional finite element model of a realizable thirty-cell beam reproduces the localized modes under the full piezoelectric physics. The closed-form prescription thereby supports tunable vibration localization, energy concentration for harvesting, and distributed sensing on a single reconfigurable structure.