A reduced-order semi-analytical model for load-modulated acoustic reflection from coated Tonpilz transducers

Underwater acoustic reconfigurable intelligent surfaces modulate the electrical load impedance of a piezoelectric transducer to control the amplitude and phase of the reflected wave. Predicting this load-dependent response is challenging: one-dimensional Mason circuits neglect oblique-incidence and shear-vertical-wave effects in elastomer coatings, while full three-dimensional finite-element models are computationally expensive and offer limited physical insight. This paper presents a reduced-order semi-analytical model for a Tonpilz-inspired tunable reflector. The model combines a primary-shear-vertical elastodynamic state-space description of a polyurethane coating, angle-dependent water loading, and a lumped electromechanical Tonpilz model terminated by an external electrical load. It yields compact expressions for the angle- and frequency-dependent acoustic reflection coefficient. Under thin-coating, moderate-incidence, and strong tangential-clamping conditions, the model shows that the reflection coefficient becomes a linear-fractional (Möbius) image of a capacitance-normalized load parameter. From this structure, physical conditions for strong load-to-reflection tracking are derived and translated into concrete reflector-design guidelines. Validation against three-dimensional finite-element simulations and tank experiments near a series-resonant frequency of 28.4 kHz shows that the model captures the dominant resonance behavior, the load-dependent amplitude and phase trends, and the angle-dependent reflection characteristics. Remaining quantitative discrepancies in absolute phase and resonance sharpness are reported explicitly and attributed to reduced-order assumptions and simplified three-dimensional structural dynamics.