field describing the sound propagation in an inhomogeneous medium: $$\frac{1}{B}\frac{\partial^2 p}{\partial t^2}- \nabla \cdot\left( \frac{1}{\rho}\nabla p\right)=0.
\label{sc_waveeq}$$ Considering sound beams with harmonic temporal dependence and Fourier expanding $B(\mathbf{r})$ and $\rho(\mathbf{r})$ since they are periodic functions with the periodicity of the lattice, equation (\[sc\_waveeq\]) can be solved by means of the plane wave expansion (PWE) method, which gives an eigenvalue problem equation:[@kushwaha_acoustic_1993; @kushwaha_theory_1994; @perez-arjona_theoretical_2007] $$\sum_{\mathbf{G}'}[\omega^2b_{\mathbf{G}-\mathbf{G}'}^{-1}-\rho_{\mathbf{G}-\mathbf{G}'}^{-1}(\mathbf{k}+\mathbf{G})\cdot(\mathbf{k}+\mathbf{G}')]p_{\mathbf{k},\mathbf{G}'}=0,
\label{sc_eigenvalue}$$ where $\omega$ is the angular frequency of the plane waves divided by the sound velocity in the host medium, $c_h$, $\mathbf{k}$ is a two-dimensional Bloch vector belonging to the irreducible Brillouin zone, $\mathbf{G}$ and $\mathbf{G}'$ are vectors of the reciprocal lattice, $p_{\mathbf{k},\mathbf{G}'}$ is the coefficient for the pressure field expanded following the Bloch-Floquet theorem

$$p(\mathbf{r})=e^{i\mathbf{k}\cdot\mathbf{r}}\sum_{\mathbf{G}'}p_{\mathbf{k},\mathbf{G}'}e^{i\mathbf{G}'\cdot\mathbf{r}},
\label{sc_bloch}$$

and $b_{\mathbf{G}-\mathbf{G}'}^{-1}$ and $\rho_{\mathbf{G}-\mathbf{G}'}^{-1}$ are the Fourier coefficients of the inverted relative values of the bulk modulus, $\bar{B}(\mathbf{r})^{-1}=B_h/B(\mathbf{r})$, and the density, $\bar{\rho}(\mathbf{r})^{-1}=\rho_h/\rho(\mathbf{r})$, respectively. Here, $B_h$ and $\rho_h$ are the bulk modulus and the density of the host medium, respectively.

In the case of a SC consisting of only two different materials, the values of the coefficients $b_{\mathbf{\bar{G}}}^{-1}$ and $\rho_{\mathbf{\bar{G}}}^{-1}$, with $\mathbf{\bar{G}}=\mathbf{G}-\mathbf{G}'$, can be found by integrating over the filled area (corresponding to steel) inside the two-dimensional unit cell. For 