Geodesic
Helmgoltz equation solutions concentrated near periodical boundary
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 27, Tome 250 (1998), pp. 83-96

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Existence of solutions of Helmholtz equation exponentially decaying away from a periodical boundary in the upper half-plane is proved. These solutions can exist for special form of the boundary under Dirichlet or Neumann boundary conditions. In both cases the boundary has a form of the resonator chain connected by narrow splits with the upper half-plane. 
@article{ZNSL_1998_250_a6,
     author = {V. Yu. Gotlib},
     title = {Helmgoltz equation solutions concentrated near periodical boundary},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {83--96},
     publisher = {mathdoc},
     volume = {250},
     year = {1998},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1998_250_a6/}
}
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V. Yu. Gotlib. Helmgoltz equation solutions concentrated near periodical boundary. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 27, Tome 250 (1998), pp. 83-96. http://geodesic.mathdoc.fr/item/ZNSL_1998_250_a6/