$r_{dL}=1.2\,\rm{mm}$, while the upper defect remains without cylinders, $r_{dU}=0$. Thus, at the output of the system, since the radii of the defects are significantly different, $\Delta k\gg\sqrt{\Omega_{UL}\Omega_{LU}}$, $\theta=0$, and the S supermode is localized in the upper waveguide while the A one is localized in the lower defect, as shown in [Fig. \[sc\_fig2\]]{}(b). Now we consider that two monochromatic sound beam sources of the same frequency are placed at the input of the system, each one in front of one of the two linear defects. If the sources are in phase, only the S supermode will be excited, sound waves will follow adiabatically the supermode along the propagation and at the output of the device there will be only sound intensity in the upper output for which $r_{dU}=0$. On the other hand, if the two sources are in opposite phase, only the A supermode will be excited and sound waves will follow it adiabatically ending up into the lower output for which $r_{dL}=1.2\,\rm{mm}$. In general, one can consider any phase difference $\varphi$ between the input sources. In this case, the normalized initial field for two sources of equal intensity can be expressed as: $$\psi_{in}=\frac{1}{\sqrt{2}}\left (\begin{array}{c}1\\ e^{i\varphi}\end{array}\right).
\label{sc_inc}$$ The S and 