Figure 2
From: Hierarchical phononic crystals for filtering multiple target frequencies of ultrasound
Relationship between bandgaps of HPC and homogenized structures. (a) Homogenized structures of HPC for different wavelengths. Depending on the wavelength, the HPC can be regarded as a different type of homogenized structure according to quasi-static homogenization theory. For example, if the wavelength is much larger than l2 but smaller than or comparable with l1, then the structure can be regarded as an l1-periodic homogenized structure comprising PDMS and a hard material with an effective density of \({\rho }_{eff}={\gamma }_{2}{\rho }_{Al}+(1-{\gamma }_{2}){\rho }_{PDMS}\) and an effective longitudinal wave modulus of \({\kappa }_{eff}={\{{\gamma }_{2}{\kappa }_{Al}^{-1}+(1-{\gamma }_{2}){\kappa }_{PDMS}^{-1}\}}^{-1}\). (b) Transmittance spectra of HPC and its homogenized structures. For each wavelength range, the stopbands of the homogenized structure accord with those of the HPC, which means that the multiple bandgaps of the hierarchical structure are the unions of bandgaps created by individual homogenized structures with periodicities corresponding to the different hierarchies.
