Singularity of the Bloch theorem in the fluid/solid phononic crystal

The physical origin that the plane wave expansion method fails to deal with the fluid/solid phononic crystals is presented. We find that the Bloch theorem is singular in some areas of this kind of system. The unphysical flat bands which appear in the band structure of the fluid/solid systems resulted from the plane wave expansion method can be gotten rid of only when the singularity is removed. As an example, an effective method is presented to calculate the band structure and transmission spectrum of the air/rigid system, in which how to correctly use the Bloch theorem in such kind of systems is shown.