Diffraction by periodic graphs

Abstract

The paper is devoted to the diffraction of graphs imbedded in periodic with respect to the action of the group We consider the Helmholtz equation

(1)

where, is the frequency of a medium, is the frequency of harmonic vibrations, and is the velocity of sound. We add to Equation (1), the transmission conditions on

where, is the set of vertices of the graph , the space of the bounded piece-wise continuous functions on with discontinuities on the space of bounded piece-wise continuous functions on with discontinuities at the vertices, and is the jump of at the point We introduce single and double layer potentials connected with the operator and reduce the transmission problem to a pseudo-differential equation on the graph We obtained necessary and sufficient conditions for the boundary pseudo-differential operators to be Fredholm in

Keywords:

• Helmholtz operators
• periodic graphs
• diffraction

AMS Subject Classifications:

• 81Q35
• 78A45

Acknowledgments

The work is partially supported by the Project CONACYT MEXICO No. 179862 and the Project SIP IPN No.20120532.