Determination of Acoustic Scattering From a Two-Dimensional Finite Phononic Crystal Using Bloch Wave Expansion

In this study the acoustic scattering is determined from a finite phononic crystal through an implementation of the Helmholtz-Kirchhoff integral theorem. The approach employs the Bloch theorem applied to a semi-infinite phononic crystal (PC) half-space. The internal pressure field of the half-space, subject to an incident acoustic monochromatic plane wave, is formulated as an expansion of the Bloch wave modes. Modal coefficients of reflected (diffracted) plane waves are arrived at via boundary condition considerations on the PC interface. Next, the PC inter-facial pressure, as determined by the Bloch wave expansion (BWE), is employed along with the Helmholtz-Kirchhoff integral equation to compute the scattered pressure from a large finite PC. Under a short wavelength limit approximation (wavelength much smaller than finite PC dimensions), the integral approach is employed to calculate the scattered pressure field for a large PC subject to an incident wave with two distinct incident angles. In two dimensions we demonstrate good agreement of scattered pressure results of large finite PC when compared against detailed finite element calculations. The work here demonstrates an efficient and accurate uniform computational framework for modeling the scattered and internal pressure fields of a large finite phononic crystal.