Geodesic
Acoustic Diffraction Problems on Periodic Graphs
Funkcionalʹnyj analiz i ego priloženiâ, Tome 48 (2014) no. 4, pp. 77-83

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We consider acoustic diffraction by graphs $\Gamma$ embedded in $\mathbb{R}^{2}$ and periodic with respect to an action of the group $\mathbb{Z}^{n}$, $n=1,2$. The diffraction problem is described by the Helmholtz equation with variable nonperiodic bounded coefficients and nonperiodic transmission conditions on the graph $\Gamma$. We introduce single and double layer potentials on $\Gamma$ generated by the Schwartz kernel of the operator inverse to the Helmholtz operator on $\mathbb{R}^{2}$ and reduce the diffraction problem to a boundary pseudodifferential equation on the graph. Necessary and sufficient conditions for the boundary operators to be Fredholm are obtained.
Keywords: Helmholtz operators, periodic graphs
Mots-clés : diffraction. 
@article{FAA_2014_48_4_a8,
     author = {V. S. Rabinovich},
     title = {Acoustic {Diffraction} {Problems} on {Periodic} {Graphs}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {77--83},
     publisher = {mathdoc},
     volume = {48},
     number = {4},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2014_48_4_a8/}
}
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V. S. Rabinovich. Acoustic Diffraction Problems on Periodic Graphs. Funkcionalʹnyj analiz i ego priloženiâ, Tome 48 (2014) no. 4, pp. 77-83. http://geodesic.mathdoc.fr/item/FAA_2014_48_4_a8/