$\rho_{\mathbf{\bar{G}}}^{-1}$ one obtains [@kushwaha_acoustic_1993; @kushwaha_theory_1994; @perez-arjona_theoretical_2007]

[\_\^[-1]{}=]{} =f+(1-f),&for $\mathbf{\bar{G}}=0$ \[sc\_coeff1\]\
,&for $\mathbf{\bar{G}}\neq0$ \[sc\_coeff2\]

where $A$ is the area of the unit cell, $f$ is the filling factor, and $\rho_c$ is the density of the filling material. In the case of a square lattice and a SC formed by cylinders of radius $r_0$, [Eq. [(\[sc\_coeff2\])]{}]{} takes the following form: $$\rho_{\mathbf{\bar{G}}}^{-1}=\left(\frac{\rho_h}{\rho_c}-1\right)2f\frac{J_1(|\mathbf{\bar{G}}|r_0)}{|\mathbf{\bar{G}}|r_0}, \,\rm{\ for\ } \mathbf{\bar{G}}\neq0,$$ where $J_1$ is the Bessel function of the first kind. The expressions for $b_{\mathbf{\bar{G}}}^{-1}$ can be obtained analogously.[@perez-arjona_theoretical_2007]

The eigenvalues of [Eq. [(\[sc\_eigenvalue\])]{}]{} can be numerically obtained and correspond to the frequencies of the allowed propagation bands in the SC. Once the frequencies are known, by using [Eq. [(\[sc\_bloch\])]{}]{} one can calculate the supermodes of the structure, i.e., the pressure field $p(\mathbf{r})$.

![(Color online) Schematic representation of the square lattice SC with (a) one linear defect and (b) two linear defects separated by a single row. The supercells are boxed with a dashed line. The SC considered in this work has a lattice constant of $a=5\,\rm{mm}$ and consists of steel cylinders of radius $r_0=2.25\,\rm{mm}$ ($\rho_c=7.8\times10^3\,\rm{kg\ m^{-3}}$ and $B_c=160\times10^9\,\rm{N\ m^{-2}}$) immersed in water ($\rho_h=10^3\,\rm{kg\ m^{-3}}$, $B_h=2.2\times10^9\,\rm{N\ m^{-2}}$).[]{data-label="sc_supercells"}](fig1.pdf){width="0.9\linewidth"}

![image](fig2.pdf){width="0.935\linewidth"}

A way to create linear defects in SCs consists in 