  The effective medium representation is fundamental in providing a
performance-to-design approach for many devices based on metamaterials. While
there are recent works in extending the effective medium concept into the
temporal domain, a direct implementation is still missing. Here, we construct
an acoustic metamaterial dynamically switching between two different
configurations with a time-varying convolution kernel, which can now
incorporate both frequency dispersion of metamaterials and temporal modulation.
We establish the effective medium formula in temporally averaging the
compressibilities, densities and even Willis coupling parameters of the two
configurations. A phase disorder between the modulation of different atoms is
found negligible on the effective medium. Our realization enables a high-level
description of metamaterials in the spatiotemporal domain, making many recent
proposals, such as magnet-free non-reciprocity, broadband slow-light and
Fresnel drag using spatiotemporal metamaterials possible for implementations in
future.
