Flexural wave reflections from time modulated boundaries

Traditional analysis of flexural wave scattering assumes that the scatter properties are time invariant and therefore the scattered wave field is at the same frequency as the incident wave and only depends on the local impedance contrast. Recent interest in time-modulated systems enables unprecedented control over wave behavior, offering opportunities for frequency and wavenumber conversions that are not possible in passive media. The control of scattered flexural waves in beams and plates with time-varying boundary conditions is of particular interest for vibration control. In this work, we present a semi-analytical model to study flexural wave reflections from time-varying boundary conditions. This model is based on Euler–Bernoulli beam theory in which expressions for reflected flexural waves are derived for various beam boundary conditions. The time dependence of the boundary conditions is incorporated by representing the solution as a Fourier series expansion to consider scattering into harmonics related to the modulation frequency and wavenumber. The semi-analytical model is compared to a realistic implementation that utilizes resonator termination with PZT patches. Numerical and experimental work is carried out to validate the model.