adding rows of cylinders with different radii to the ones forming the square lattice. If we introduce one linear defect composed of cylinders of radius $r_d$ (see [Fig. \[sc\_supercells\]]{}(a)), bands corresponding to the modes of the individual waveguide are obtained into a frequency band gap in the projected band diagram of the SC, allowing for sound guiding inside the SC. Similarly, if we introduce two parallel linear defects with radii $r_{dU}$ (for the upper one) and $r_{dL}$ (for the lower one) separated, for example, by one row of cylinders of radius $r_0$ (see [Fig. \[sc\_supercells\]]{}(b)), bands corresponding to different supermodes of the system appear into the band gap of the projected band diagram. For the calculation of the projected band diagrams of the structures with linear defects it is necessary to change the unit cell of the SC to a supercell, which contains several of the previous unit cells in the $y$ direction, as shown in [Fig. \[sc\_supercells\]]{}. The PWE method can also be used in this case.[@vasseur_absolute_2008] However, $b_{\mathbf{\bar{G}}}^{-1}$ and $\rho_{\mathbf{\bar{G}}}^{-1}$ have to be recalculated for the corresponding supercell with [Eq. [(\[sc\_coeff2\])]{}]{}. For the case of one linear defect in a supercell containing nine cylinders ([Fig. \[sc\_supercells\]]{}(a)), [Eq. [(\[sc\_coeff2\])]{}]{} 