Local gaps in three-dimensional periodic media

The propagation of acoustic waves in a medium with a periodic array of small inclusions of arbitrary shape is considered. The inclusion size <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>a</mml:mi> </mml:math> is much smaller than the array period. We show that global gaps do not exist if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>a</mml:mi> </mml:math> is small enough. We introduce and study the concept of local gaps, which depends on the choice of the wave vector <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mstyle displaystyle="true" scriptlevel="0"> <mml:mrow> <mml:mi mathvariant="bold-italic">k</mml:mi> </mml:mrow> </mml:mstyle> </mml:math> . We analytically determine the location of local gaps for the Dirichlet and transmission problems.