Abstract
The design of acoustic metamaterials requires accurate modeling of the dynamics present at all relevant time and length scales. Recent work has shown that the effective dynamic properties of inhomogeneous elastic materials can result in constitutive relations that couple strain to momentum and velocity to stress, which is often referred to as Willis coupling [Willis, Wave Motion, 3(1), 1-11, (1981)]. The current work will examine macroscale acoustic propagation for waves on different time scales in a material by solving the coupled first-order equations of motion with the constitutive relations that account for Willis coupling. Specifically, second-order perturbation theory will be used to examine the classic problem of a high-frequency, low-amplitude “signal” wave superposed on a low-frequency, high-amplitude “pump” wave. Of particular interest is the slowly changing momentum bias generated by the pump wave and implications on dynamic control of signal wave propagation. Analysis and discussion will be restricted to one-dimensional wave motion.
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