A space-adiabatic theorem for longitudinal and transversal wave motion analysis of graded metamaterials

The use of graded metamaterials is widespread in the fields of wave manipulation and energy harvesting. However, they have the drawback of causing energy scattering after a certain period due to wave reflection or trapping, known as the rainbow effect, in some cases. To address this issue, we extend the adiabatic theorem to the spatial modulation field to mitigate the impact of the rainbow effect, thereby enhancing transmission at the interface of graded metamaterials. Firstly, an analytical expression for the limiting gradient in the space-adiabatic theorem is derived. Then, it is applied to study the longitudinal wave motion in a spring-mass with graded resonators (SMGR) system and the transversal wave motion in a beam with graded resonators (BGR) system. The numerical study demonstrates that if the spatial gradient of system properties is small enough to satisfy the adiabatic theorem, wave scattering can almost be neglected, and transmission will experience a significant increase compared to the non-adiabatic case. Finally, the critical boundary between adiabatic and non-adiabatic cases is determined using the fitting method. This study introduces a design strategy for graded metamaterials, offering opportunities for communication, sensing, energy harvesting, and various other applications.