Effective medium theory of the one-dimensional resonance phononic crystal

A general theoretical scheme to describe the effective modulus and mass density for acoustic metamaterials is presented. For such a purpose, an effective medium theory of a one-dimensional acoustic waveguide containing subwavelength-sized Helmholtz resonators is formulated. It is shown that, when the wavelength is much larger than the periodic length and the size of the resonators, the whole composite structure can be treated as an effective homogeneous medium in accounting for its acoustic properties. It is also shown that the acoustic characteristics, such as the effective modulus and the effective mass density, can be determined precisely from the transmission and the reflection data. The calculated effective modulus and effective mass density confirm that this structure behaves as a homogeneous metamaterial with a negative effective modulus in a frequency range just above the resonant frequency.